1 Introduction
By Dirichlet’s approximation theorem, for any real number
$\unicode[STIX]{x1D709}$
and any positive integer
$N$
,
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert n\unicode[STIX]{x1D709}\Vert \leqslant \frac{1}{N+1},\end{eqnarray}$$
where
$\Vert \cdot \Vert$
denotes the distance to the nearest integer. Hardy and Littlewood [Reference Hardy and Littlewood22] conjectured a similar result for the distances
$\Vert n^{k}\unicode[STIX]{x1D709}\Vert$
(with given exponent
$k\in \mathbb{N}$
). Vinogradov [Reference Vinogradov54] proved the following.
Theorem 1.1. Let
$\unicode[STIX]{x1D709}\in \mathbb{R}$
be a given real number and
$N\in \mathbb{N}$
a given positive integer. Then, for every
$k\in \mathbb{N}$
, there exists an exponent
$\unicode[STIX]{x1D702}_{k}>0$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert \unicode[STIX]{x1D709}n^{k}\Vert \ll _{k}N^{-\unicode[STIX]{x1D702}_{k}}.\end{eqnarray}$$
Throughout this paper we use the standard notation
$\ll$
, where the index denotes the dependence of the implicit constant; furthermore, we use instead of
$\ll$
sometimes the
${\mathcal{O}}$
-notation.
In the case of squares, Heilbronn [Reference Heilbronn24] improved Vinogradov’s exponent to
$-\frac{1}{2}+\unicode[STIX]{x1D700}$
(with arbitrary
$\unicode[STIX]{x1D700}>0$
). For the related literature up to 1986 we refer to the classical monograph of Baker [Reference Baker1]. The best-known exponent in the quadratic case is
$-\frac{4}{7}+\unicode[STIX]{x1D700}$
due to Zaharescu [Reference Zaharescu59]. However, his method is not applicable to higher powers. It is an open conjecture that
$\unicode[STIX]{x1D702}_{k}$
can be taken as
$1-\unicode[STIX]{x1D700}$
with arbitrary
$\unicode[STIX]{x1D700}>0$
.
Generalizations to arbitrary polynomials
$f\in \mathbb{Z}[X]$
with
$f(0)=0$
are due to Davenport [Reference Davenport17] and Cook [Reference Cook13]. Let
$2\leqslant k_{1}\leqslant \cdots \leqslant k_{s}$
be integers,
$\unicode[STIX]{x1D709}_{i}\in \mathbb{R}$
for
$1\leqslant i\leqslant s$
and
$\ell =2^{1-k_{1}}+\cdots +2^{1-k_{s}}$
. Then Cook [Reference Cook15] also showed that
$\Vert \unicode[STIX]{x1D709}_{1}n_{1}^{k_{1}}+\cdots +\unicode[STIX]{x1D709}_{s}n_{s}^{k_{s}}\Vert \ll _{k_{1},\ldots ,k_{s},\unicode[STIX]{x1D700}}N^{-s/(s+(1-\ell )2^{k_{s}-1})+\unicode[STIX]{x1D700}}$
. Wooley [Reference Wooley57] considered the Diophantine inequality over smooth numbers to obtain an improvement. The proofs of these results are based on a sophisticated treatment of the occurring exponential sums. In a recent paper, Lê and Spencer [Reference Lê and Spencer29] proved the following.
Theorem 1.2 [Reference Lê and Spencer29, Theorem 3].
Let
$N\in \mathbb{N}$
and
$h\in \mathbb{Z}[X]$
be a polynomial with integer coefficients such that for every non-zero integer
$q$
there exists a solution
$n$
to the congruence
$h(n)\equiv 0~\text{mod}\,q$
. Then there is an exponent
$\unicode[STIX]{x1D702}>0$
depending only on the degree of
$h$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert \unicode[STIX]{x1D709}h(n)\Vert \ll _{h}N^{-\unicode[STIX]{x1D702}}\end{eqnarray}$$
for arbitrary
$\unicode[STIX]{x1D709}\in \mathbb{R}$
.
A recent proof of a related well-known conjecture concerning the Vinogradov integral is also due to Wooley [Reference Wooley58] for
$k=3$
and to Bourgain et al [Reference Bourgain, Demeter and Guth12] in the general case. Baker [Reference Baker2] used this approach to improve on Diophantine inequalities as considered in Theorem 1.2.
Danicic [Reference Danicic16] considered two-dimensional extensions of the above problem and showed that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\max (\Vert \unicode[STIX]{x1D6FC}n^{2}\Vert ,\Vert \unicode[STIX]{x1D6FD}n^{2}\Vert )\ll N^{-\unicode[STIX]{x1D702}}\end{eqnarray}$$
uniformly in
$N$
,
$\unicode[STIX]{x1D6FC}$
and
$\unicode[STIX]{x1D6FD}$
. Higher powers were investigated by Liu [Reference Liu31]. Cook [Reference Cook13, Reference Cook14] generalized these results to a system of polynomials without constant term.
Definition 1.1. Let
$h_{1},\ldots ,h_{k}$
be a system of polynomials in
$\mathbb{Z}[X]$
. This system is called jointly intersective if, for every
$q\neq 0$
, there exists an
$n\in \mathbb{Z}$
such that
$h_{i}(n)\equiv 0~(\text{mod}\,q)$
for
$i=1,\ldots ,k$
.
Remark 1.3. The concept of jointly intersective polynomials was introduced independently (under different names) in different areas by Bergelson et al [Reference Bergelson, Leibman and Lesigne7], Lê [Reference Lê28], Rice [Reference Rice45] and Wierdl [Reference Wierdl56].
Note that the common root condition in Definition 1.1 is necessary, which is shown by a simple counterexample in the case
$\unicode[STIX]{x1D709}=a/q$
.
Now we link the concept of jointly intersective polynomials with intersective sets. In the case
$k=1$
an arbitrary subset
${\mathcal{I}}\subset \mathbb{Z}\setminus \{0\}$
is called intersective if
$$\begin{eqnarray}{\mathcal{I}}\cap (S-S)\neq \emptyset\end{eqnarray}$$
whenever the upper density of
$S\subset \mathbb{Z}$
is positive. A famous result, independently established by Furstenberg [Reference Furstenberg19] and Sárközy [Reference Sárközy47], states that the set of
$k$
th powers and the set of shifted primes
$p+1$
and
$p-1$
are intersective sets. Moreover, a sufficient condition for a set being intersective is due to Kamae and Mendès France [Reference Kamae and Mendès France25] and was later extended by Nair [Reference Nair41–Reference Nair43]. Lê and Spencer [Reference Lê and Spencer29] established the following.
Theorem 1.4 [Reference Lê and Spencer29, Theorem 4].
Let
$\ell$
be a positive integer,
$h_{1},\ldots ,h_{k}$
jointly intersective polynomials,
$A=(a_{ij})$
an arbitrary
$\ell \times k$
matrix with real entries and
$N\in \mathbb{N}$
. Then there is an exponent
$\unicode[STIX]{x1D702}>0$
depending only on
$\ell$
and on the polynomials
$h_{i}$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\max _{1\leqslant i\leqslant \ell }\mathop{\biggl\|\mathop{\sum }_{j=1}^{k}a_{ij}h_{j}(n)\biggr\|\ll }\nolimits_{\ell ,h_{1},\ldots ,h_{k}}N^{-\unicode[STIX]{x1D702}},\end{eqnarray}$$
where the bound is uniform in
$A$
.
Harman [Reference Harman23] considered the sequence
$\unicode[STIX]{x1D6FC}p^{k}$
for
$\unicode[STIX]{x1D6FC}>0$
and
$k$
a positive integer, where
$p$
runs through the prime numbers. Baker and Kolesnik [Reference Baker and Kolesnik4] considered the distribution modulo one of the more general sequence
$\unicode[STIX]{x1D6FC}p^{\unicode[STIX]{x1D703}}$
. Improvements of the latter for the case
$\unicode[STIX]{x1D703}=1$
have been established by Matomäki [Reference Matomäki34]. Recent refinements of the statement were given by Baker [Reference Baker3]. Motivated by the above observations concerning intersective sets, Lê and Spencer [Reference Lê and Spencer30] also proved an extension of Theorem 1.4 to polynomials evaluated at prime numbers.
The aim of our paper is to establish such Diophantine inequalities for the Piatetski–Shapiro sequence and for pseudo-polynomial sequences, for instance for the sequence
$n^{c}+n^{k}$
with
$c>0$
a non-integral real number and
$k\in \mathbb{N}$
. In §2 we introduce the concept of van der Corput sets and we formulate our results in detail. In §3 we prove that also in the multivariate setting every van der Corput set is a Heilbronn set. This extends a classical one-dimensional result of Montgomery [Reference Montgomery38]. Section 4 is devoted to exponential sum estimates and the final §§6 and 7 deal with single and multiple pseudo-polynomials, respectively.
2 Van der Corput sets and statement of results
In the following we introduce van der Corput sets for multiparameter systems, that is, for
$\mathbb{Z}^{k}$
-actions (in the terminology of ergodic theory). For more details, see Bergelson and Lesigne [Reference Bergelson and Lesigne8].
Definition 2.1. A subset
${\mathcal{H}}\subset \mathbb{Z}^{k}\setminus \{\mathbf{0}\}$
is a van der Corput set (vdC set) if for any family
$(u_{\mathbf{n}})_{\mathbf{n}\in \mathbb{Z}^{k}}$
of complex numbers of modulus 1 such that
$$\begin{eqnarray}\text{for all }\mathbf{h}\in {\mathcal{H}},\quad \lim _{N_{1},\ldots ,N_{k}\rightarrow \infty }\frac{1}{N_{1}\cdots N_{k}}\mathop{\sum }_{0\leqslant \mathbf{n}<(N_{1},\ldots ,N_{k})}u_{\mathbf{n}+\mathbf{h}}\overline{u_{\mathbf{n}}}=0,\end{eqnarray}$$
we have
$$\begin{eqnarray}\lim _{N_{1},\ldots ,N_{k}\rightarrow \infty }\frac{1}{N_{1}\cdots N_{k}}\mathop{\sum }_{0\leqslant \mathbf{n}<(N_{1},\ldots ,N_{k})}u_{\mathbf{n}}=0.\end{eqnarray}$$
Here in the limit
$N_{1},\ldots ,N_{k}$
tend to infinity independently and
${<}$
stands for the product order.
Equivalently,
${\mathcal{H}}$
is a vdC set if any family
$(x_{\mathbf{n}})_{\mathbf{n}\in \mathbb{N}^{k}}$
of real numbers having the property that for all
$\mathbf{h}\in {\mathcal{H}}$
the family
$(x_{\mathbf{n}+\mathbf{h}}-x_{\mathbf{n}})_{\mathbf{n}\in \mathbb{N}^{k}}$
is uniformly distributed mod 1, is itself uniformly distributed mod 1. The concept of uniform distribution for multiparameter systems (so-called multisequences) was investigated in various papers; see for instance Losert and Tichy [Reference Losert and Tichy32], Kirschenhofer and Tichy [Reference Kirschenhofer and Tichy26], Tichy and Zeiner [Reference Tichy and Zeiner52] and the book of Drmota and Tichy [Reference Drmota and Tichy18].
Van der Corput’s difference theorem states that in the case
$k=1$
the full set
$\mathbb{N}$
of positive integers is a van der Corput set (cf. [Reference Kuipers and Niederreiter27, Theorem 3.1]). However, this is only a sufficient condition. Therefore, the question of the necessary “size” of vdC sets arises. For various aspects of van der Corput’s difference theorem, we refer to the recent paper of Bergelson and Moreira [Reference Bergelson and Moreira9]. Delange observed that also the sets
$q\mathbb{N}$
, where
$q\geqslant 2$
is an integer, are van der Corput sets. More general examples like the
$k$
th powers or shifted primes
$p+1$
and
$p-1$
are due to Kamae and Mendès France [Reference Kamae and Mendès France25]. In particular, they proved in the case
$k=1$
that each van der Corput set is also intersective. The converse does not hold true, as was shown by Bourgain [Reference Bourgain11].
In his seminal paper, Ruzsa [Reference Ruzsa46] gave four equivalent definitions of vdC sets. We refer the interested reader to Ch. 2 of Montgomery [Reference Montgomery38] or the important work of Bergelson and Lesigne [Reference Bergelson and Lesigne8] for a detailed account of vdC sets.
Definition 2.2. Let
${\mathcal{H}}\in \mathbb{Z}^{k}\setminus \{\mathbf{0}\}$
. We call
${\mathcal{H}}$
a Heilbronn set if for every
$\unicode[STIX]{x1D709}\in \mathbb{R}^{k}$
and every
$\unicode[STIX]{x1D700}>0$
there is an
$\mathbf{h}\in {\mathcal{H}}$
such that
$$\begin{eqnarray}\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert <\unicode[STIX]{x1D700},\end{eqnarray}$$
where
$\cdot$
denotes the standard inner product.
Following Montgomery, we want to analyze quantitative aspects of Heilbronn sets. This is related to the spectral definition of van der Corput sets given by Kamae and Mendés France [Reference Kamae and Mendès France25] and its multidimensional variant by Bergelson and Lesigne [Reference Bergelson and Lesigne8]. For each subset
${\mathcal{H}}\subset \mathbb{Z}^{k}\setminus \{\mathbf{0}\}$
, we denote by
${\mathcal{T}}={\mathcal{T}}({\mathcal{H}})$
the set of real trigonometric polynomials
$$\begin{eqnarray}T(\mathbf{x})=a_{0}+\mathop{\sum }_{\mathbf{h}\in {\mathcal{H}}}a_{\mathbf{h}}\cos (2\unicode[STIX]{x1D70B}\mathbf{h}\cdot \mathbf{x})\end{eqnarray}$$
with
$T(\mathbf{x})\geqslant 0$
for all
$\mathbf{x}\in \mathbb{R}^{k}$
and
$T(\mathbf{0})=1$
. Furthermore, we set
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}({\mathcal{H}}):=\inf _{T\in {\mathcal{T}}({\mathcal{H}})}a_{0}.\end{eqnarray}$$
In order to provide a quantitative result on Heilbronn sets, we introduce the following quantity:
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}({\mathcal{H}})=\sup _{\unicode[STIX]{x1D709}\in \mathbb{R}^{k}}\inf _{\mathbf{h}\in {\mathcal{H}}}\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert .\end{eqnarray}$$
Then our first result is the following.
Theorem 2.1. Let
${\mathcal{H}}\subset \mathbb{Z}^{k}\setminus \{0\}$
.
(1)
${\mathcal{H}}$
is a van der Corput set if and only if
$\unicode[STIX]{x1D6FF}({\mathcal{H}})=0$
.(2)
${\mathcal{H}}$
is a Heilbronn set if and only if
$\unicode[STIX]{x1D6FE}({\mathcal{H}})=0$
.(3)
$\unicode[STIX]{x1D6FE}({\mathcal{H}})\leqslant \unicode[STIX]{x1D6FF}({\mathcal{H}})$
.(4) Any van der Corput set is a Heilbronn set.
Remark 2.2. In the one-dimensional case the “if” statement of Assertion (1) was already proved by Kamae and Mendés France [Reference Kamae and Mendès France25] and the “only if” part is due to Ruzsa [Reference Ruzsa46]. The general case was shown by Bergelson and Lesigne [Reference Bergelson and Lesigne8]. Assertion (2) is obvious. The one-dimensional version of Assertion (3) was already shown by Montgomery [Reference Montgomery38]. In §3, we prove the multidimensional case of Assertion (3). Finally, Assertion (4) is a direct consequence of Assertion (3).
In Montgomery [Reference Montgomery38], one can also find counterexamples which show that in general the converse is false. Former results by Bergelson, Boshernitzan, Kolesnik, Lesigne, Madritsch, Quas, Son, Tichy and Wierdl provide us with many examples of vdC sets.
∙ Let
$f,g\in \mathbb{Z}[X]$
be such that for any
$q\in \mathbb{N}$
there exists
$n\in \mathbb{N}$
such that
$q$
divides
$f$
and
$g$
. Then the two-dimensional set
$\{(f(n),g(n)):n\in \mathbb{N}\}$
is a vdC set (see [Reference Bergelson and Lesigne8]).∙ Let
$f,g\in \mathbb{Z}[X]$
be polynomials with zero constant term. Then the two-dimensional set
$\{(f(p-1),g(p-1)):p\text{ prime}\}$
is a vdC set (see [Reference Bergelson and Lesigne8]).∙ Let
$c>1$
be irrational and
$b\neq 0$
. Then the set
$\{\lfloor bn^{c}\rfloor :n\in \mathbb{N}\}$
is a vdC set (see [Reference Boshernitzan, Kolesnik, Quas and Wierdl10]).∙ Let
$b,d\neq 0$
be such that
$b/d$
is irrational,
$c\geqslant 1$
,
$a>0$
and
$a\neq c$
. Then the set
$\{\lfloor bn^{c}+dn^{a}\rfloor :n\in \mathbb{N}\}$
is a vdC set (see [Reference Boshernitzan, Kolesnik, Quas and Wierdl10]).∙ Let
$b\neq 0$
,
$c>1$
be irrational and
$d$
any real number. Then the set
$\{\lfloor bn^{c}(\log n)^{d}\rfloor :n\in \mathbb{N}\}$
is a vdC set (see [Reference Boshernitzan, Kolesnik, Quas and Wierdl10]).∙ Let
$b\neq 0$
,
$c>1$
be rational and
$d\neq 0$
. Then the set
$\{\lfloor bn^{c}(\log n)^{d}\rfloor :n\in \mathbb{N}\}$
is a vdC set (see [Reference Boshernitzan, Kolesnik, Quas and Wierdl10]).∙ Let
$b,d\neq 0$
,
$c\geqslant 1$
and
$a>1$
. Then the set
$\{\lfloor bn^{c}+d(\log n)^{a}\rfloor :n\in \mathbb{N}\}$
is a vdC set (see [Reference Boshernitzan, Kolesnik, Quas and Wierdl10]).∙ Let
$\unicode[STIX]{x1D6FC}_{i}$
be positive integers and
$\unicode[STIX]{x1D6FD}_{i}$
be positive non-integral reals. Then the
$(k+\ell )$
-dimensional sets and$$\begin{eqnarray}\{((p-1)^{\unicode[STIX]{x1D6FC}_{1}},\ldots ,(p-1)^{\unicode[STIX]{x1D6FC}_{k}},\lfloor (p-1)^{\unicode[STIX]{x1D6FD}_{1}}\rfloor ,\ldots ,\lfloor (p-1)^{\unicode[STIX]{x1D6FD}_{\ell }}\rfloor ):p\text{ prime}\}\end{eqnarray}$$
are vdC sets (see [Reference Bergelson, Kolesnik, Madritsch, Son and Tichy5] and [Reference Madritsch and Tichy33]).$$\begin{eqnarray}\{((p+1)^{\unicode[STIX]{x1D6FC}_{1}},\ldots ,(p+1)^{\unicode[STIX]{x1D6FC}_{k}},\lfloor (p+1)^{\unicode[STIX]{x1D6FD}_{1}}\rfloor ,\ldots ,\lfloor (p+1)^{\unicode[STIX]{x1D6FD}_{\ell }}\rfloor ):p\text{ prime}\}\end{eqnarray}$$
Furthermore, Bergelson et al [Reference Bergelson, Kolesnik and Son6] showed under some mild conditions that for a function
$f$
from a Hardy field the sequence
$(\{f(p)\})_{p\text{ prime}}$
is uniformly distributed mod 1. From this, they deduced general classes of vdC sets and by our Theorem 2.1 they are Heilbronn sets, too.
In §§6 and 7, we want to show that for sets of the form
$\{\lfloor n^{c}\rfloor +n^{k}:n\in \mathbb{N}\}$
, where
$c>1$
is not an integer and
$k\in \mathbb{N}$
, and multidimensional variants thereof, we may replace
$\unicode[STIX]{x1D700}$
by some negative power
$N^{-\unicode[STIX]{x1D702}}$
depending only on the exponents
$c$
and
$k$
. Therefore, we want to introduce the concept of pseudo-polynomials.
Definition 2.3. Let
$\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\ldots ,\unicode[STIX]{x1D6FC}_{d},\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2},\ldots ,\unicode[STIX]{x1D703}_{d}$
be positive reals such that
$1\leqslant \unicode[STIX]{x1D703}_{1}<\cdots <\unicode[STIX]{x1D703}_{d}$
and at least one
$\unicode[STIX]{x1D703}_{j}\not \in \mathbb{Z}$
for
$1\leqslant j\leqslant d$
. A function
$f:\mathbb{R}\rightarrow \mathbb{R}$
of the form
$$\begin{eqnarray}f(x)=\unicode[STIX]{x1D6FC}_{1}x^{\unicode[STIX]{x1D703}_{1}}+\cdots +\unicode[STIX]{x1D6FC}_{d}x^{\unicode[STIX]{x1D703}_{d}}\end{eqnarray}$$
is called a pseudo-polynomial.
By abuse of notation, we write
$\deg f=\unicode[STIX]{x1D703}_{d}$
.
We investigate Diophantine inequalities for sequences of the form
$(\lfloor f(n)\rfloor )_{n\geqslant 1}$
. A simple example is the Piatetski–Shapiro-type sequence
$(\lfloor n^{c}\rfloor +n^{k})_{n\geqslant 1}$
with
$c>1$
a non-integral real and
$k\in \mathbb{N}$
.
Theorem 2.3. Let
$\unicode[STIX]{x1D709}$
be a real number,
$N\in \mathbb{N}$
sufficiently large and
$f$
a pseudo-polynomial. Then there exists an exponent
$\unicode[STIX]{x1D702}>0$
depending only on
$f$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert \unicode[STIX]{x1D709}\lfloor f(n)\rfloor \Vert \ll _{f}N^{-\unicode[STIX]{x1D702}}.\end{eqnarray}$$
When applying our result to the sequence
$(\lfloor n^{c}\rfloor +n^{k})_{n\geqslant 1}$
, we obtain the following.
Corollary 2.4. Let
$\unicode[STIX]{x1D709}$
be real,
$c>1$
a non-integral real number,
$N\in \mathbb{N}$
sufficiently large and
$k\in \mathbb{N}$
. Then, for arbitrary
$\unicode[STIX]{x1D700}>0$
,
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert \unicode[STIX]{x1D709}\lfloor n^{c}+n^{k}\rfloor \Vert \ll \left\{\begin{array}{@{}ll@{}}N^{-(1/2)(1/(2^{\lceil c\rceil +1}-1))+\unicode[STIX]{x1D700}}\quad & \text{if }c>k,\\ N^{-(1/(2^{k-1}(k+2)))+\unicode[STIX]{x1D700}}\quad & \text{if }c<k.\end{array}\right.\end{eqnarray}$$
Remark 2.5. Using the methods of Mauduit and Rivat [Reference Mauduit and Rivat35, Reference Mauduit and Rivat36] and of Morgenbesser [Reference Morgenbesser39] combined with Spiegelhofer [Reference Spiegelhofer51] and Müllner and Spiegelhofer [Reference Müllner and Spiegelhofer40], who considered Beatty sequences, one could improve this exponent in the case
$c>k$
.
Similarly to the results above, we may consider sequences over the primes.
Theorem 2.6. Let
$\unicode[STIX]{x1D709}$
be a real number,
$N\in \mathbb{N}$
sufficiently large and
$f$
a pseudo-polynomial. Then there exists an exponent
$\unicode[STIX]{x1D702}>0$
depending only on
$f$
such that
$$\begin{eqnarray}\min _{\substack{ 1\leqslant p\leqslant N \\ p\text{ prime}}}\Vert \unicode[STIX]{x1D709}\lfloor f(p)\rfloor \Vert \ll _{f}N^{-\unicode[STIX]{x1D702}}.\end{eqnarray}$$
Corollary 2.7. Let
$\unicode[STIX]{x1D709}$
be real,
$c>1$
a non-integral real number,
$N\in \mathbb{N}$
sufficiently large and
$k\in \mathbb{N}$
. Then, for arbitrary
$\unicode[STIX]{x1D700}>0$
,
$$\begin{eqnarray}\min _{\substack{ 1\leqslant p\leqslant N \\ p\text{ prime}}}\Vert \unicode[STIX]{x1D709}\lfloor p^{c}+p^{k}\rfloor \Vert \ll \left\{\begin{array}{@{}ll@{}}N^{-(1/2)(1/(2^{\lceil c\rceil +1}-1))+\unicode[STIX]{x1D700}}\quad & \text{if }c>k,\\ N^{-(1/4^{k-1}(k+2))+\unicode[STIX]{x1D700}}\quad & \text{if }c<k.\end{array}\right.\end{eqnarray}$$
Finally, we state multidimensional variants of these estimates.
Theorem 2.8. Let
$\ell$
be a positive integer,
$f_{1},\ldots ,f_{k}$
pseudo-polynomials of pairwise different degree,
$A=(a_{ij})$
an arbitrary
$\ell \times k$
matrix with real entries and
$N\in \mathbb{N}$
. Then there is an exponent
$\unicode[STIX]{x1D702}>0$
depending only on
$\ell$
and on the polynomials
$f_{i}$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\max _{1\leqslant i\leqslant \ell }\mathop{\biggl\|\mathop{\sum }_{j=1}^{k}a_{ij}\lfloor f_{j}(n)\rfloor \biggr\|\ll }\nolimits_{\ell ,f_{1},\ldots ,f_{k}}N^{-\unicode[STIX]{x1D702}},\end{eqnarray}$$
where the bound is uniform in
$A$
.
Again we may consider sequences of integer parts of pseudo-polynomials over the primes.
Theorem 2.9. Let
$\ell$
be a positive integer,
$f_{1},\ldots ,f_{k}$
pseudo-polynomials of pairwise different degree,
$A=(a_{ij})$
an arbitrary
$\ell \times k$
matrix with real entries and
$N\in \mathbb{N}$
. Then there is an exponent
$\unicode[STIX]{x1D702}>0$
depending only on
$\ell$
and on the polynomials
$f_{i}$
such that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\max _{1\leqslant i\leqslant \ell }\mathop{\biggl\|\mathop{\sum }_{j=1}^{k}a_{ij}\lfloor f_{j}(p)\rfloor \biggr\|\ll }\nolimits_{\ell ,f_{1},\ldots ,f_{k}}N^{-\unicode[STIX]{x1D702}},\end{eqnarray}$$
where the bound is uniform in
$A$
.
3 Every van der Corput set is also a Heilbronn set
We first present the following easy proof that any van der Corput set is a Heilbronn set. Let
${\mathcal{H}}\subset \mathbb{Z}^{k}\setminus \{0\}$
be a van der Corput set. It is an easy deduction that
${\mathcal{H}}$
must be intersective in the multidimensional sense. Then for a given
$\unicode[STIX]{x1D709}\in \mathbb{R}^{k}$
we consider the set
$$\begin{eqnarray}A=\{\mathbf{h}\in \mathbb{Z}^{k}:\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert <\unicode[STIX]{x1D700}/2\}.\end{eqnarray}$$
Since
${\mathcal{H}}$
is intersective and
$A$
has positive density, we deduce that
$$\begin{eqnarray}{\mathcal{H}}\cap (A-A)\neq \emptyset ,\end{eqnarray}$$
which proves Theorem 2.1.
A quantitative analysis of van der Corput and Heilbronn sets can be provided using the parameter
$\unicode[STIX]{x1D6FF}$
introduced in §2. Let
$N$
be a positive real. Then for a fixed van der Corput set
${\mathcal{H}}$
we set
${\mathcal{H}}_{N}:={\mathcal{H}}\cap [-N,N]^{d}$
. Then
$\unicode[STIX]{x1D6FF}({\mathcal{H}}_{N})$
tends to
$0$
as
$N$
tends to infinity. Only a few upper bounds for this quantity are known. In particular, for the set of shifted primes Slijepčević [Reference Slijepčević50] could show that
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\{p-1\leqslant N:p\text{ prime}\})\ll (\log N)^{-1+o(1)}.\end{eqnarray}$$
Similar results are known for squares [Reference Slijepčević49]. In case of Heilbronn sets, one has to consider the parameter
$\unicode[STIX]{x1D6FE}$
introduced in §2 instead of
$\unicode[STIX]{x1D6FF}$
.
The following proposition immediately yields a complete proof of Theorem 2.1.
Proposition 3.1. Let
${\mathcal{H}}\in \mathbb{Z}^{k}\setminus \{\mathbf{0}\}$
. Then
$\unicode[STIX]{x1D6FE}({\mathcal{H}})\leqslant \unicode[STIX]{x1D6FF}({\mathcal{H}})$
.
Proof. Our proof follows Montgomery’s proof for
$k=1$
in [Reference Montgomery38]. Let
$T\in {\mathcal{T}}({\mathcal{H}})$
as above, let
$0<\unicode[STIX]{x1D700}\leqslant \frac{1}{2}$
and set
$f(x)=\max (0,1-\Vert x\Vert /\unicode[STIX]{x1D700})$
. Then we consider
$$\begin{eqnarray}g(\unicode[STIX]{x1D709}):=a_{0}+\mathop{\sum }_{\mathbf{h}\in {\mathcal{H}}}a_{\mathbf{h}}f(\mathbf{h}\cdot \unicode[STIX]{x1D709}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D709}\in \mathbb{R}^{k}$
. Since
$f$
is continuous and of bounded variation, its Fourier transform converges absolutely to
$f$
. Thus,
$$\begin{eqnarray}g(\unicode[STIX]{x1D709})=a_{0}+\mathop{\sum }_{\mathbf{h}\in {\mathcal{H}}}a_{\mathbf{h}}\mathop{\sum }_{m\in \mathbb{Z}}\widehat{f}(m)e(m\mathbf{h}\cdot \unicode[STIX]{x1D709}).\end{eqnarray}$$
The function
$f$
is even. Hence, its Fourier coefficients
$\widehat{f}(m)$
are real. Moreover,
$g(\unicode[STIX]{x1D709})$
is real and hence
$$\begin{eqnarray}g(\unicode[STIX]{x1D709})=a_{0}+\mathop{\sum }_{\mathbf{h}\in {\mathcal{H}}}a_{\mathbf{h}}\mathop{\sum }_{m\in \mathbb{Z}}\widehat{f}(m)\cos (2\unicode[STIX]{x1D70B}m\mathbf{h}\cdot \unicode[STIX]{x1D709}).\end{eqnarray}$$
Inverting the order of summation yields
$$\begin{eqnarray}g(\unicode[STIX]{x1D709})=\mathop{\sum }_{m\in \mathbb{Z}}\widehat{f}(m)T(m\unicode[STIX]{x1D709}).\end{eqnarray}$$
A simple calculation shows that
$$\begin{eqnarray}\widehat{f}(m)=\frac{1}{\unicode[STIX]{x1D700}}\biggl(\frac{\sin (\unicode[STIX]{x1D70B}m\unicode[STIX]{x1D700})}{\unicode[STIX]{x1D70B}m}\biggr)\geqslant 0\quad \text{and}\quad T(m\unicode[STIX]{x1D703})\geqslant 0\quad \text{ for all }m.\end{eqnarray}$$
Thus,
$g(\unicode[STIX]{x1D709})$
is greater than the contribution of the term for
$m=0$
in the above sum. Since
$\widehat{f}(0)=\unicode[STIX]{x1D700}$
and
$T(0)=1$
, we get
$$\begin{eqnarray}g(\unicode[STIX]{x1D709})\geqslant \unicode[STIX]{x1D700}.\end{eqnarray}$$
Now, if
$\unicode[STIX]{x1D700}>a_{0}$
, then there must be at least one
$\mathbf{h}\in {\mathcal{H}}$
such that
$a_{\mathbf{h}}>0$
and
$f(\mathbf{h}\cdot \unicode[STIX]{x1D709})>0$
. Hence,
$\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert <\unicode[STIX]{x1D700}$
. Since this holds for every
$\unicode[STIX]{x1D700}>a_{0}$
, we obtain
$\inf _{\mathbf{h}\in {\mathcal{H}}}\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert \leqslant a_{0}$
. Furthermore, since this holds for every polynomial
$T\in {\mathcal{T}}({\mathcal{H}})$
, we get
$\inf _{\mathbf{h}\in {\mathcal{H}}}\Vert \mathbf{h}\cdot \unicode[STIX]{x1D709}\Vert \leqslant \unicode[STIX]{x1D6FF}$
. Finally, since this holds for every
$\unicode[STIX]{x1D709}\in \mathbb{R}^{k}$
, it follows that
$\unicode[STIX]{x1D6FE}\leqslant \unicode[STIX]{x1D6FF}$
, which proves the proposition.◻
Remark 3.2. Applying the results of Slijepčević [Reference Slijepčević49, Reference Slijepčević50], Proposition 3.1 implies upper bounds for
$\unicode[STIX]{x1D6FE}$
in case of shifted primes and squares.
4 Exponential sum estimates for the case
$\unicode[STIX]{x1D703}_{r}>k$
Before stating the proofs of the main theorems, we collect some well-known facts on exponential sums which will occur in the sequel. Let
$f$
be a pseudo-polynomial. Then there exist a real function
$g$
and a polynomial
$P$
such that
$f(x)=g(x)+P(x)$
,
$g(x)=\sum _{j=1}^{r}\unicode[STIX]{x1D6FC}_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$1<\unicode[STIX]{x1D703}_{1}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\not \in \mathbb{Z}$
for
$j=1,\ldots ,r$
. Let
$k$
be the degree of
$P$
and we set
$k=0$
if
$P\equiv 0$
. By abuse of notation, we write
$$\begin{eqnarray}\deg f=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D703}_{r}\quad & \text{if }\unicode[STIX]{x1D703}_{r}>k,\\ k\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
We only consider the one-dimensional case, since the multidimensional case is similar. The proof is by supposing that
$\Vert \unicode[STIX]{x1D709}\lfloor f(n)\rfloor \Vert >M^{-1}$
for every
$1\leqslant n\leqslant N$
. Then using Vinogradov’s method we approximate the indicator function and by our assumption this yields a lower bound for an exponential sum of the form
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant n\leqslant N}e(\unicode[STIX]{x1D6FD}f(n)),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FD}>0$
is some real number.
The aim of this section is to provide upper bounds for this exponential sum that subsequentially violate the lower bounds, yielding a contradiction. In the proof we use different arguments but essentially the same exponential sum estimate if
$\unicode[STIX]{x1D6FD}$
is large or
$\unicode[STIX]{x1D6FD}$
is small, respectively, where these sizes have to be understood modulo one. In both cases we use Weyl differencing, which is equivalent to derivation. If
$g$
is the dominant part of
$f$
, this means that
$\deg f\not \in \mathbb{Z}$
; we may differentiate as often as we wish till the resulting function has the desired behavior. On the other hand, if the polynomial part
$P$
is dominant, we cannot differentiate that freely since after
$k$
steps we lose the polynomial and therefore the dominant part. Therefore, we have to further consider two cases for
$k>\unicode[STIX]{x1D703}_{r}$
or not.
Let
$\unicode[STIX]{x1D70C}$
be a real satisfying
$$\begin{eqnarray}0<\unicode[STIX]{x1D70C}<\frac{1}{\lfloor \deg f\rfloor +3}.\end{eqnarray}$$
First, we assume that
$$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-\deg f}<|\unicode[STIX]{x1D6FD}|\leqslant N^{1/10}.\end{eqnarray}$$
Now we distinguish the cases of
$\unicode[STIX]{x1D703}_{r}>k$
and
$\unicode[STIX]{x1D703}_{r}<k$
.
∙ If
$\unicode[STIX]{x1D703}_{r}>k$
, then
$\deg f=\unicode[STIX]{x1D703}_{r}$
and the tools in §4 allow us to deal with the exponential sums in the whole range $$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}<|\unicode[STIX]{x1D6FD}|\leqslant N^{1/10}.\end{eqnarray}$$
∙ If
$k>\unicode[STIX]{x1D703}_{r}$
, then
$\deg f=k$
. Since
$\unicode[STIX]{x1D70C}-\deg f<\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}$
, the estimates of §4 are not sufficient and we need further tools from §5 to deal with the additional range $$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-k}<|\unicode[STIX]{x1D6FD}|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}.\end{eqnarray}$$
Secondly, if
$$\begin{eqnarray}0<|\unicode[STIX]{x1D6FD}|\leqslant N^{\unicode[STIX]{x1D70C}-\deg f},\end{eqnarray}$$
then a different argument allows us to reuse the estimates for bigger
$\unicode[STIX]{x1D6FD}$
. Furthermore, we note that the exponent
$1/10$
is an artifact of [Reference Bergelson, Kolesnik, Madritsch, Son and Tichy5, Lemma 2.3] which we use in the proof.
The following lemma will be the main tool for the range
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}<|\unicode[STIX]{x1D6FD}|<N^{1/10}$
.
Lemma 4.1 [Reference Bergelson, Kolesnik, Madritsch, Son and Tichy5, Lemma 2.5].
Let
$X,k,q\in \mathbb{N}$
with
$k,q\geqslant 0$
and set
$K=2^{k}$
and
$Q=2^{q}$
. Let
$P(x)$
be a polynomial of degree
$k$
with real coefficients. Let
$g(x)$
be a real
$(q+k+2)$
-times continuously differentiable function on
$[X,2X]$
such that
$|g^{(r)}(x)|\asymp GX^{-r}$
$(r=1,\ldots ,q+k+2)$
. Then, if
$G=o(X^{q+2})$
for
$G$
and
$X$
large enough, we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{X<n\leqslant 2X}e(g(n)+P(n)) & \ll & \displaystyle X^{1-1/K}+X\biggl(\frac{\log ^{k}X}{G}\biggr)^{1/K}\nonumber\\ \displaystyle & & \displaystyle +\,X\biggl(\frac{G}{X^{q+2}}\biggr)^{1/(4KQ-2K)},\nonumber\end{eqnarray}$$
where
$A\asymp B$
means that
$A$
is of the same order as
$B$
, i.e.
$A\ll B$
and
$B\ll A$
.
In particular, we get that the
$G$
in the lemma above is of the right size if
$\biggl|\unicode[STIX]{x1D6FD}\biggr|$
lies in the first range.
Proposition 4.2. Let
$P(x)$
be a polynomial of degree
$k$
and
$g(x)=\sum _{j=1}^{r}d_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$r\geqslant 1$
,
$d_{r}\neq 0$
,
$d_{j}$
real,
$0<\unicode[STIX]{x1D703}_{1}<\unicode[STIX]{x1D703}_{2}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\notin \mathbb{N}$
. Assume that
$\ell <\unicode[STIX]{x1D703}_{r}<\ell +1$
for some integer
$\ell$
. Let
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\leqslant |\unicode[STIX]{x1D709}|\leqslant N^{1/10}$
. Then, for arbitrary
$\unicode[STIX]{x1D700}>0$
,
$$\begin{eqnarray}\biggl|\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D709}g(n)+\unicode[STIX]{x1D709}P(n))\biggr|\ll N^{1-1/(8KL-4K)},\end{eqnarray}$$
where
$K=2^{k}$
and
$L=2^{\ell }$
.
Proof. We split the sum into
${\leqslant}\log N$
subsums of the form
$$\begin{eqnarray}\mathop{\sum }_{X\leqslant n\leqslant 2X}e(\unicode[STIX]{x1D709}g(n)+\unicode[STIX]{x1D709}P(n))\end{eqnarray}$$
and denote by
$S$
a typical one of them. Because of the polynomial structure of
$g$
and since
$\unicode[STIX]{x1D703}_{r}\not \in \mathbb{Z}$
, we get for
$j\geqslant 1$
that
$$\begin{eqnarray}|g^{(j)}(x)|\asymp X^{\unicode[STIX]{x1D703}_{r}-j}.\end{eqnarray}$$
Thus, an application of Lemma 4.1 with
$q=\ell$
yields
$$\begin{eqnarray}S\ll X^{1-1/K}+X\biggl(\frac{\log ^{k}X}{|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}}}\biggr)^{1/K}+X\biggl(\frac{|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}}}{X^{\ell +2}}\biggr)^{1/(4KL-2K)}.\end{eqnarray}$$
Summing over all subsums and using the above bounds on
$\biggl|\unicode[STIX]{x1D709}\biggr|$
, we get
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D709}g(n)+\unicode[STIX]{x1D709}P(n)) & \ll & \displaystyle N(|\unicode[STIX]{x1D709}|N^{\unicode[STIX]{x1D703}_{r}})^{-1/K}+N^{1-1/(8KL-4K)}\nonumber\\ \displaystyle & \ll & \displaystyle N^{1-1/(8KL-4K)},\nonumber\end{eqnarray}$$
which yields the desired bound.◻
Before we will prove the corresponding estimate for sums over primes, we need some standard tools. The first one is von Mangoldt’s function defined by
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(n)=\left\{\begin{array}{@{}ll@{}}\log p\quad & \text{if }n=p^{k}\text{ for some prime }p\text{ and an integer }k\geqslant 1,\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
Lemma 4.3 [Reference Mauduit and Rivat37, Lemma 11].
Let
$g$
be a function such that
$\biggl|g(n)\biggr|\leqslant 1$
for all integers
$n$
. Then
$$\begin{eqnarray}\biggl|\mathop{\sum }_{p\leqslant P}g(p)\biggr|\ll \frac{1}{\log P}\max _{t\leqslant P}\biggl|\mathop{\sum }_{n\leqslant t}\unicode[STIX]{x1D6EC}(n)g(n)\biggr|+\sqrt{P}.\end{eqnarray}$$
Next we introduce the
$s$
-fold divisor sum
$d_{s}(n)$
, i.e.
$$\begin{eqnarray}d_{s}(n)=\mathop{\sum }_{x_{1}\cdots x_{s}=n}1.\end{eqnarray}$$
The central tool in the treatment of the exponential sum over primes is Vaughan’s identity, which we use to subdivide the weighted exponential sum into several sums of Types I and II.
Lemma 4.4 [Reference Bergelson, Kolesnik, Madritsch, Son and Tichy5, Lemma 2.3].
Assume
$F(x)$
to be any function defined on the real line, supported on
$[X,2X]$
and bounded by
$F_{0}$
. Let further
$U,V,Z$
be any parameters satisfying
$3\leqslant U<V<Z<2X$
,
$Z\geqslant 4U^{2}$
,
$X\geqslant 32Z^{2}U$
,
$V^{3}\geqslant 32X$
and
$Z-\frac{1}{2}\in \mathbb{N}$
. Then
$$\begin{eqnarray}\biggl|\mathop{\sum }_{X<n\leqslant 2X}\unicode[STIX]{x1D6EC}(n)F(n)\biggr|\ll K\log P+F_{0}+L(\log X)^{8},\end{eqnarray}$$
where
$K$
and
$L$
are defined by
$$\begin{eqnarray}\displaystyle K & = & \displaystyle \max _{M}\mathop{\sum }_{m=1}^{\infty }d_{3}(m)\biggl|\mathop{\sum }_{Z<n\leqslant M}F(mn)\biggr|\quad \text{(Type I)},\nonumber\\ \displaystyle L & = & \displaystyle \sup \mathop{\sum }_{m=1}^{\infty }d_{4}(m)\biggl|\mathop{\sum }_{U<n<V}b(n)F(mn)\biggr|\quad \text{(Type II)},\nonumber\end{eqnarray}$$
where the supremum is taken over all arithmetic functions
$b(n)$
satisfying
$|b(n)|\leqslant d_{3}(n).$
Combining Vaughan’s identity with Lemma 4.1 provides the analogous estimate for the sum over primes.
Proposition 4.5. Let
$P(x)$
be a polynomial of degree
$k$
and
$g(x)=\sum _{j=1}^{r}d_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$r\geqslant 1$
,
$d_{r}\neq 0$
,
$d_{j}$
real,
$0<\unicode[STIX]{x1D703}_{1}<\unicode[STIX]{x1D703}_{2}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\notin \mathbb{N}$
. Assume that
$\ell <\unicode[STIX]{x1D703}_{r}<\ell +1$
for some integer
$\ell$
. Let
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\leqslant |\unicode[STIX]{x1D709}|\leqslant N^{1/10}$
. Then, for arbitrary
$\unicode[STIX]{x1D700}>0$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D709}g(p)+\unicode[STIX]{x1D709}P(p))\biggr|\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{1-\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D700}}+N^{1-2/3K+\unicode[STIX]{x1D700}}+N^{1-\unicode[STIX]{x1D70C}/K+\unicode[STIX]{x1D700}}+N^{1-1/(64KL^{5}-4K)+\unicode[STIX]{x1D700}},\nonumber\end{eqnarray}$$
where
$K=2^{k}$
and
$L=2^{\ell }$
.
Proof. We start with an application of Lemma 4.3 which transforms the sum over the primes into the weighted sum
$$\begin{eqnarray}\biggl|\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D709}g(p)+\unicode[STIX]{x1D709}P(p))\biggr|\ll \frac{1}{\log N}\max \biggl|\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D6EC}(n)e(\unicode[STIX]{x1D709}(g(n)+P(n)))\biggr|+N^{1/2}.\end{eqnarray}$$
Then we split the inner sum into
${\leqslant}\log N$
subsums of the form
$$\begin{eqnarray}\biggl|\mathop{\sum }_{X<n\leqslant 2X}\unicode[STIX]{x1D6EC}(n)e(\unicode[STIX]{x1D709}(g(n)+P(n)))\biggr|\end{eqnarray}$$
with
$2X\leqslant N$
and we denote by
$S$
a typical one of them. We may assume that
$X\geqslant N^{1-\unicode[STIX]{x1D70C}}$
.
Applying Vaughan’s identity (Lemma 4.4) with the parameters
$U=\frac{1}{4}X^{1/5}$
,
$V=4X^{1/3}$
and
$Z$
, the unique number in
$\frac{1}{2}+\mathbb{N}$
which is closest to
$\frac{1}{4}X^{2/5}$
yields
$$\begin{eqnarray}S\ll 1+(\log X)S_{1}+(\log X)^{8}S_{2},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle S_{1} & = & \displaystyle \mathop{\sum }_{x<2X/Z}d_{3}(x)\mathop{\sum }_{y>Z,X/x<y<2X/x}e(\unicode[STIX]{x1D709}(g(xy)+P(xy))),\nonumber\\ \displaystyle S_{2} & = & \displaystyle \mathop{\sum }_{X/V<x\leqslant 2X/U}d_{4}(x)\mathop{\sum }_{U<y<V,X/x<y\leqslant 2X/x}b(y)e(\unicode[STIX]{x1D709}(g(xy)+P(xy))).\nonumber\end{eqnarray}$$
We start estimating
$S_{1}$
. Since
$d_{3}(x)\ll x^{\unicode[STIX]{x1D700}}$
, we have
$$\begin{eqnarray}|S_{1}|\ll X^{\unicode[STIX]{x1D700}}\mathop{\sum }_{x\leqslant 2X/Z}\biggl|\mathop{\sum }_{\substack{ X/x<y(2X/x) \\ y>Z}}e(\unicode[STIX]{x1D709}(g(xy)+P(xy)))\biggr|.\end{eqnarray}$$
We fix
$x$
and write
$Y=X/x$
for short. Since
$\unicode[STIX]{x1D703}_{r}\not \in \mathbb{Z}$
, we obtain for
$j\geqslant 1$
that
$$\begin{eqnarray}\biggl|\frac{\unicode[STIX]{x2202}^{j}g(xy)}{\unicode[STIX]{x2202}y^{j}}\biggr|\asymp X^{\unicode[STIX]{x1D703}_{r}}Y^{-j}.\end{eqnarray}$$
For
$j\geqslant 5(\ell +1)$
, we get
$$\begin{eqnarray}\unicode[STIX]{x1D709}X^{\unicode[STIX]{x1D703}_{r}}Y^{-j}\leqslant N^{1/10}X^{\unicode[STIX]{x1D703}_{r}-(2/5)j}\ll X^{-1/2}.\end{eqnarray}$$
Thus, an application of Lemma 4.1 yields the following estimate:
$$\begin{eqnarray}\displaystyle |S_{1}| & \ll & \displaystyle X^{\unicode[STIX]{x1D700}}\mathop{\sum }_{x\leqslant 2X/Z}Y[Y^{-1/K}+(\log Y)^{k}(|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}})^{-1/K}+X^{-(1/2)1/(4K\cdot 8L^{5}-2K)}]\nonumber\\ \displaystyle & \ll & \displaystyle X^{1+\unicode[STIX]{x1D700}}(\log X)[X^{-2/5K}+X^{-\unicode[STIX]{x1D70C}/K}+X^{-1/(64KL^{5}-4K)}],\end{eqnarray}$$
where we have used that
$k/K<1$
and
$\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D703}_{r}+1)<\unicode[STIX]{x1D70C}(\lfloor \deg f\rfloor +2)<1$
by (4.1).
Now we turn our attention to the second sum
$S_{2}$
. We split the range
$(X/V,2X/U]$
into
${\leqslant}\log X$
subintervals of the form
$(X_{1},2X_{1}]$
. Thus,
$$\begin{eqnarray}|S_{2}|\ll (\log X)X^{\unicode[STIX]{x1D700}}\mathop{\sum }_{X_{1}<x\leqslant 2X_{1}}\biggl|\mathop{\sum }_{\substack{ U<y<V \\ X/x<y\leqslant 2X/x}}b(y)e(\unicode[STIX]{x1D709}(g(xy)+P(xy)))\biggr|.\end{eqnarray}$$
An application of Cauchy’s inequality together with the estimate
$\biggl|b(y)\biggr|\ll X^{\unicode[STIX]{x1D700}}$
yields
$$\begin{eqnarray}\displaystyle |S_{2}|^{2} & \ll & \displaystyle (\log X)^{2}X^{2\unicode[STIX]{x1D700}}X_{1}\mathop{\sum }_{X_{1}<x\leqslant 2X_{1}}\biggl|\mathop{\sum }_{\substack{ U<y<V \\ X/x<y\leqslant 2X/x}}b(y)e(\unicode[STIX]{x1D709}(g(xy)+P(xy)))\biggr|^{2}\nonumber\\ \displaystyle & \ll & \displaystyle (\log X)^{2}X^{4\unicode[STIX]{x1D700}}X_{1}\biggl(X_{1}\frac{X}{X_{1}}\nonumber\\ \displaystyle & & \displaystyle +\,\biggl|\mathop{\sum }_{X_{1}<x\leqslant 2X_{1}}\mathop{\sum }_{A<y_{1}<y_{2}\leqslant B}\!e(\unicode[STIX]{x1D709}(g(xy_{1})-g(xy_{2})+P(xy_{1})-P(xy_{2})))\biggr|\biggr),\nonumber\end{eqnarray}$$
where
$A=\max \{U,X/x\}$
and
$B=\min \{U,2X/x\}$
. A change of the order of summation yields
$$\begin{eqnarray}\displaystyle & & \displaystyle |S_{2}|^{2}\ll (\log X)^{2}X^{4\unicode[STIX]{x1D700}}X_{1}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \biggl(\!X\!+\!\!\mathop{\sum }_{A<y_{1}<y_{2}\leqslant B}\biggl|\mathop{\sum }_{X_{1}<x\leqslant 2X_{1}}\!\!e(\unicode[STIX]{x1D709}(g(xy_{1})-g(xy_{2})+P(xy_{1})-P(xy_{2})))\biggr|\biggr).\nonumber\end{eqnarray}$$
We fix
$y_{1}$
and
$y_{2}\neq y_{1}$
. Similarly as above, we get
$$\begin{eqnarray}\biggl|\frac{\unicode[STIX]{x2202}^{j}(g(xy_{1})-g(xy_{2})+P(xy_{1})-P(xy_{2}))}{\unicode[STIX]{x2202}x^{j}}\biggr|\asymp \frac{|y_{1}-y_{2}|}{y_{1}}X^{\unicode[STIX]{x1D703}_{r}}X_{1}^{-j}.\end{eqnarray}$$
In this case we suppose that
$j\geqslant 2\lfloor \unicode[STIX]{x1D703}_{r}\rfloor +3$
in order to obtain
$$\begin{eqnarray}\unicode[STIX]{x1D709}\frac{|y_{1}-y_{2}|}{y_{1}}X^{\unicode[STIX]{x1D703}_{r}}X_{1}^{-j}\ll X^{1/10+\unicode[STIX]{x1D703}_{r}}\biggl(\frac{X}{V}\biggr)^{-j}\ll X^{1/10+\unicode[STIX]{x1D703}_{r}-(2/3)j}\ll X^{-1/2}.\end{eqnarray}$$
Thus, again an application of Lemma 4.1 yields
$$\begin{eqnarray}\displaystyle |S_{2}|^{2} & \ll & \displaystyle (\log X)^{2}X^{4\unicode[STIX]{x1D700}}X_{1}\biggl(X+\mathop{\sum }_{A<y_{1}<y_{2}\leqslant B}X_{1}(X_{1}^{-1/K}+(|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}})^{-1/K}\nonumber\\ \displaystyle & & \displaystyle +\,X^{-(1/2)(1/(4K\cdot 2L^{2}-2K))})\biggr)\nonumber\\ \displaystyle & \ll & \displaystyle (\log X)^{2}X^{4\unicode[STIX]{x1D700}}(X^{2-2/3K}+X^{2-\unicode[STIX]{x1D70C}/K}+X^{2-1/(16KL^{2}-4K)}).\end{eqnarray}$$
Inserting the two estimates (4.3) and (4.4) into (4.2) together with a summation over all intervals proves the proposition.◻
5 Exponential sum estimates for the case
$\unicode[STIX]{x1D703}_{r}<k$
If
$k>\unicode[STIX]{x1D703}_{r}$
, the estimates in Propositions 4.2 and 4.5 are not covering the whole range
$N^{\unicode[STIX]{x1D70C}-\deg f}<\biggl|\unicode[STIX]{x1D6FD}\biggr|\leqslant N^{1/10}$
, but only
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}<\biggl|\unicode[STIX]{x1D6FD}\biggr|N^{1/10}$
. In this section we develop estimates for the missing range
$$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-k}<|\unicode[STIX]{x1D6FD}|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D70C}$
is as in (4.1).
As above, we write
$f(x)=g(x)+P(x)$
, where
$P$
is a polynomial of degree
$k$
with real coefficients and
$g(x)=\sum _{j=1}^{r}\unicode[STIX]{x1D6FC}_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$1<\unicode[STIX]{x1D703}_{1}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\not \in \mathbb{Z}$
for
$j=1,\ldots ,r$
. The idea is to separate the polynomial
$P$
and the real function
$g$
by means of a partial summation. Then for the sum over the polynomial we use standard estimates due to Weyl [Reference Weyl55] and Harman [Reference Harman23] for the integer and prime cases, respectively. Since these are standard methods, we present only the non-standard steps and refer the interested reader to Ch. 3 of Montgomery [Reference Montgomery38], Ch. 4 of Nathanson [Reference Nathanson44] or the monograph of Graham and Kolesnik [Reference Graham and Kolesnik21] for a more complete account on the method of Weyl and van der Corput.
Proposition 5.1. Let
$f$
be a pseudo-polynomial and
$\unicode[STIX]{x1D700}>0$
. Suppose that
$\unicode[STIX]{x1D70C}(k+3)<1$
and that
$$\begin{eqnarray}N^{3\unicode[STIX]{x1D70C}-k}\leqslant |\unicode[STIX]{x1D709}|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}.\end{eqnarray}$$
Then, for sufficiently large
$N$
,
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D709}f(n))\ll N^{1-\unicode[STIX]{x1D70C}2^{1-k}+\unicode[STIX]{x1D700}}.\end{eqnarray}$$
Before proving this first result, we need some tools and definitions. We start with the forward difference operator
$\unicode[STIX]{x1D6E5}_{d}$
, which is the linear operator defined on functions
$f$
by the formula
$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{d}(f)(x)=f(x+d)-f(x).\end{eqnarray}$$
For
$\ell \geqslant 2$
, we define the iterated difference operator
$\unicode[STIX]{x1D6E5}_{d_{\ell },d_{\ell -1},\ldots ,d_{1}}$
by
$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{d_{\ell },d_{\ell -1},\ldots ,d_{1}}=\unicode[STIX]{x1D6E5}_{d_{\ell }}\circ \unicode[STIX]{x1D6E5}_{d_{\ell -1},\ldots ,d_{1}}=\unicode[STIX]{x1D6E5}_{d_{\ell }}\circ \unicode[STIX]{x1D6E5}_{d_{\ell -1}}\circ \cdots \circ \unicode[STIX]{x1D6E5}_{d_{1}}.\end{eqnarray}$$
The following lemma describes the idea of “Weyl differencing”.
Lemma 5.2 [Reference Nathanson44, Lemma 4.13].
Let
$N_{1}$
,
$N_{2}$
,
$N$
and
$\ell$
be integers such that
$\ell \geqslant 1$
,
$N_{1}<N_{2}$
and
$N_{2}-N_{1}\leqslant N$
. Let
$f(n)$
be a real-valued arithmetic function. Then
$$\begin{eqnarray}\biggl|\mathop{\sum }_{n=N_{1}+1}^{N_{2}}e(f(n))\biggr|^{2^{\ell }}\!\leqslant (2N)^{2^{\ell }-\ell -1}\mathop{\sum }_{|d_{1}|<N}\cdots \mathop{\sum }_{|d_{\ell }|<N}\mathop{\sum }_{n\in I(d_{\ell },\ldots ,d_{1})}\!e(\unicode[STIX]{x1D6E5}_{d_{\ell },\ldots ,d_{1}}(f)(n)),\end{eqnarray}$$
where
$I(d_{\ell },\ldots ,d_{1})$
is an interval of consecutive integers contained in
$[N_{1}+1,N_{2}]$
.
In order to estimate the innermost sum of the previous lemma, we use the following observation.
Lemma 5.3 [Reference Nathanson44, Lemma 4.7].
For every real number
$\unicode[STIX]{x1D6FC}$
and all integers
$N_{1}<N_{2}$
,
$$\begin{eqnarray}\mathop{\sum }_{n=N_{1}+1}^{N_{2}}e(\unicode[STIX]{x1D6FC}n)\ll \min (N_{2}-N_{1},\Vert \unicode[STIX]{x1D6FC}\Vert ^{-1}).\end{eqnarray}$$
The final tool is the following estimate for sums of minima.
Lemma 5.4 [Reference Nathanson44, Lemma 4.9].
Let
$\unicode[STIX]{x1D6FC}$
be a real number. If
$$\begin{eqnarray}\biggl|\unicode[STIX]{x1D6FC}-\frac{a}{b}\biggr|\leqslant \frac{1}{b^{2}},\end{eqnarray}$$
where
$b\geqslant 1$
and
$(a,b)=1$
, then for any negative real numbers
$H$
and
$N$
we have
$$\begin{eqnarray}\mathop{\sum }_{h=1}^{H}\min \biggl(N,\frac{1}{\Vert \unicode[STIX]{x1D6FC}h\Vert }\biggr)\ll \biggl(b+H+N+\frac{HN}{b}\biggr)\max \{1,\log b\}.\end{eqnarray}$$
The following lemma links the estimate of the exponential sum with the leading coefficient of the polynomial part and the non-polynomial part.
Lemma 5.5. Let
$X,k\in \mathbb{N}$
with
$k\geqslant 0$
and set
$K=2^{k-1}$
. Let
$P$
be a polynomial of degree
$k$
with real coefficients and let
$\unicode[STIX]{x1D6FC}$
be the leading coefficient. Let
$g(x)$
be a real
$k$
-times continuously differentiable function on
$[X,2X]$
such that
$\biggl|g^{(r)}(x)\biggr|\asymp GX^{-r}~(r=1,\ldots ,k)$
. Then, for
$G$
and
$X$
large enough, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{X<n\leqslant 2X}e(P(n)+g(n))\biggr|^{2^{k-1}}\nonumber\\ \displaystyle & & \displaystyle \qquad \ll X^{2^{k-1}-1}+(1+G)X^{2^{k-1}-k+\unicode[STIX]{x1D700}}\mathop{\sum }_{t=1}^{k!X^{k-1}}\min \biggl(X,\frac{1}{\Vert t\unicode[STIX]{x1D6FC}\Vert }\biggr)\nonumber\end{eqnarray}$$
with arbitrary
$\unicode[STIX]{x1D700}>0$
.
Proof. Our first tool is Lemma 5.2 with
$\ell =k-1$
to get
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{X<n\leqslant 2X}e(P(n)+g(n))\biggr|^{2^{k-1}}\nonumber\\ \displaystyle & & \displaystyle \quad \ll X^{2^{k-1}-1}+X^{2^{k-1}-k}\mathop{\sum }_{1\leqslant |d_{1}|<X}\nonumber\\ \displaystyle & & \displaystyle \qquad \cdots \mathop{\sum }_{1\leqslant |d_{k-1}|<X}\mathop{\sum }_{n\in I(d_{k-1},\ldots ,d_{1})}e(\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(P+g)(n)).\nonumber\end{eqnarray}$$
Now we want to separate the polynomial and non-polynomial parts. To this end, we set
$$\begin{eqnarray}a_{n}=e(\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}P(n))\quad \text{and}\quad b_{n}=e(\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}g(n)).\end{eqnarray}$$
Then an application of partial summation yields
$$\begin{eqnarray}\mathop{\sum }_{X<n\leqslant 2X}a_{n}b_{n}\leqslant \biggl|\mathop{\sum }_{X<n\leqslant 2X}a_{n}\biggr|+\mathop{\sum }_{X<h\leqslant 2X}|b_{h}-b_{h+1}|\biggl|\mathop{\sum }_{X<n\leqslant h}a_{n}\biggr|.\end{eqnarray}$$
For the non-polynomial part
$b_{h}-b_{h+1}$
, we note the following representation for the forward difference operator (cf. [Reference Graham and Kolesnik21, Lemma 2.7]):
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(g)(n)\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{0}^{1}\cdots \int _{0}^{1}\frac{\unicode[STIX]{x2202}^{k-1}}{\unicode[STIX]{x2202}t_{1}\cdots \unicode[STIX]{x2202}t_{k-1}}g(n+t_{1}d_{1}+\cdots +t_{k-1}d_{k-1})\,\text{d}t_{1}\cdots \text{d}t_{k-1}.\nonumber\end{eqnarray}$$
Together with the mean value theorem, we get
$$\begin{eqnarray}|b_{h}-b_{h+1}|\asymp X^{k-1}GX^{-k}=GX^{-1}.\end{eqnarray}$$
Now we turn our attention to the polynomial part, which means the sum of
$a_{n}$
. Since
$\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(P)(n)=k!d_{1}\cdots d_{k-1}\unicode[STIX]{x1D6FC}n+m(d_{1},\ldots ,d_{k-1})$
, where
$m$
is a function not depending on
$n$
, an application of Lemma 5.3 yields
$$\begin{eqnarray}\biggl|\mathop{\sum }_{X<n\leqslant h}e(\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}P(n))\biggr|\ll \min \biggl(h,\frac{1}{\Vert k!d_{1}\cdots d_{k-1}\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}\Vert }\biggr).\end{eqnarray}$$
Putting the two estimates together, we get
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{X<n\leqslant 2X}e(P(n)+g(n))\biggr|^{2^{k-1}}\nonumber\\ \displaystyle & & \displaystyle \quad \ll X^{2^{k-1}-1}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,X^{2^{k-1}-k}\mathop{\sum }_{1\leqslant |d_{1}|<X}\cdots \mathop{\sum }_{1\leqslant |d_{k-1}|<X}(1+G)\min \biggl(X,\frac{1}{\Vert k!d_{1}\cdots d_{k-1}\unicode[STIX]{x1D6FC}\Vert }\biggr).\nonumber\end{eqnarray}$$
Noting that for every
$t\leqslant k!X^{k-1}$
there are
$X^{\unicode[STIX]{x1D700}}$
solutions
$d_{1},\ldots ,d_{k-1}$
to
$$\begin{eqnarray}t=k!d_{1}\ldots d_{k-1},\end{eqnarray}$$
we get that
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{X<n\leqslant 2X}e(P(n)+g(n))\biggr|^{2^{k-1}}\nonumber\\ \displaystyle & & \displaystyle \qquad \ll X^{2^{k-1}-1}+(1+G)X^{2^{k-1}-k+\unicode[STIX]{x1D700}}\mathop{\sum }_{t=1}^{k!X^{k-1}}\min \biggl(X,\frac{1}{\Vert t\unicode[STIX]{x1D6FC}\Vert }\biggr),\nonumber\end{eqnarray}$$
which proves the lemma.◻
We have seen that the estimates reduce to an approximation of the leading coefficient of the polynomial part of
$f$
. The following lemma shows that the leading coefficient of
$\unicode[STIX]{x1D709}P(x)$
can always be approximated well provided that
$\unicode[STIX]{x1D709}$
is in the “medium” range.
Lemma 5.6. Let
$N$
,
$\unicode[STIX]{x1D6FC}$
,
$\unicode[STIX]{x1D70C}$
and
$\unicode[STIX]{x1D709}$
be positive reals and let
$k\geqslant 2$
be a positive integer. Suppose that
$\unicode[STIX]{x1D70C}<(k+3)^{-1}$
and that
$$\begin{eqnarray}N^{3\unicode[STIX]{x1D70C}-k}<|\unicode[STIX]{x1D709}|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}.\end{eqnarray}$$
Then there exist coprime
$a,b\in \mathbb{Z}$
such that
$$\begin{eqnarray}|b\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}-a|\leqslant b^{-2}\quad \text{and}\quad N^{2\unicode[STIX]{x1D70C}}\leqslant b\leqslant N^{k-2\unicode[STIX]{x1D70C}}\end{eqnarray}$$
provided that
$N$
is sufficiently large.
Proof. By Dirichlet’s approximation theorem, there exist coprime
$a,b\in \mathbb{Z}$
such that
$$\begin{eqnarray}|b\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}-a|\leqslant N^{-k+2\unicode[STIX]{x1D70C}}\quad \text{and}\quad 1\leqslant b\leqslant N^{k-2\unicode[STIX]{x1D70C}}.\end{eqnarray}$$
If
$b\geqslant N^{2\unicode[STIX]{x1D70C}}$
, then there is nothing to show. Suppose the contrary. We distinguish different cases for the size of
$b$
.
- Case 1.
- $2\leqslant b<N^{2\unicode[STIX]{x1D70C}}$
. In this case we get $$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\gg |\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}|\geqslant \biggl|\frac{a}{b}\biggr|-\frac{1}{b^{2}}\geqslant \frac{1}{2b}\geqslant \frac{1}{2}N^{-2\unicode[STIX]{x1D70C}}.\end{eqnarray}$$
Since
$3\unicode[STIX]{x1D70C}<1<\unicode[STIX]{x1D703}_{r}$
, this contradicts our lower bound for sufficiently large
$N$
. - Case 2.
- $b=1$
. This case requires a further distinction according to whether
$a=0$
or not.
- Case 2.1.
- $\biggl|\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}\biggr|\geqslant \frac{1}{2}$
. It follows that $$\begin{eqnarray}N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\gg |\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}|\geqslant {\textstyle \frac{1}{2}},\end{eqnarray}$$
which is absurd for
$N$
sufficiently large. - Case 2.2.
- $\biggl|\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}\biggr|<\frac{1}{2}$
. This implies that
$a=0$
, which yields $$\begin{eqnarray}N^{3\unicode[STIX]{x1D70C}-k}\ll |\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}|\leqslant N^{-k+2\unicode[STIX]{x1D70C}},\end{eqnarray}$$
which again is absurd for sufficiently large
$N$
.◻
Now we have all tools at hand to prove the estimate in the integer case.
Proof of Proposition 5.1.
We divide the sum into
${\leqslant}(\log N)$
subsums of the form
$$\begin{eqnarray}\mathop{\sum }_{X<n\leqslant 2X}e(\unicode[STIX]{x1D709}f(n))\end{eqnarray}$$
and denote by
$S$
a typical one of them. Without loss of generality, we may suppose that
$X\geqslant N^{1-\unicode[STIX]{x1D70C}}$
. For
$\ell \geqslant 1$
, we have
$$\begin{eqnarray}|\unicode[STIX]{x1D709}g^{(\ell )}(n)|\asymp |\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}-\ell }.\end{eqnarray}$$
Thus, an application of Lemma 5.5 yields
$$\begin{eqnarray}S^{K}\ll X^{K-1}+(1+|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}})X^{K-k+\unicode[STIX]{x1D700}}\mathop{\sum }_{t=1}^{k!X^{k-1}}\min \biggl(X,\frac{1}{\Vert t\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}\Vert }\biggr).\end{eqnarray}$$
Let
$a,b$
be positive integers such that
$$\begin{eqnarray}\biggl|\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}-\frac{a}{b}\biggr|\leqslant \frac{1}{b^{2}}.\end{eqnarray}$$
Using Lemma 5.4, we get for the sum
$$\begin{eqnarray}S^{K}\ll X^{K-1}+(1+|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}})X^{K-k+\unicode[STIX]{x1D700}}\biggl(b+\frac{X^{k}}{b}\biggr).\end{eqnarray}$$
Taking the
$K$
th root and summing over all dyadic intervals
$[X,2X]$
, we get
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D709}f(n))\ll N^{1-1/K+\unicode[STIX]{x1D700}}+N^{\unicode[STIX]{x1D70C}/K}N^{1-k/K+\unicode[STIX]{x1D700}}\biggl(b+\frac{N^{k}}{b}\biggr)^{1/K},\end{eqnarray}$$
where we used that
$\biggl|\unicode[STIX]{x1D709}\biggr|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}$
. Finally, we derive the bounds for
$b$
of Lemma 5.6,
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D709}f(n))\ll N^{1-1/K+\unicode[STIX]{x1D700}}+N^{\unicode[STIX]{x1D70C}/K}N^{1+\unicode[STIX]{x1D700}}N^{-2\unicode[STIX]{x1D70C}/K}\ll N^{1-\unicode[STIX]{x1D70C}/K}+\unicode[STIX]{x1D700},\end{eqnarray}$$
which is the desired result.◻
Now we turn our attention to the prime case. Therefore, we will reuse the central idea of the good approximation but we have to adopt the Weyl differencing part. In particular, we will obtain bilinear forms (sums over products) instead of the usual exponential sums.
Proposition 5.7. Let
$N$
and
$\unicode[STIX]{x1D70C}$
be positive reals and
$f$
be a pseudo-polynomial. If
$\unicode[STIX]{x1D70C}(k+3)<1$
and
$\unicode[STIX]{x1D709}$
is such that
$$\begin{eqnarray}N^{3\unicode[STIX]{x1D70C}-k}<|\unicode[STIX]{x1D709}|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\end{eqnarray}$$
holds, then
$$\begin{eqnarray}\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D709}f(p))\ll N^{1-\unicode[STIX]{x1D70C}4^{1-k}+\unicode[STIX]{x1D700}}\end{eqnarray}$$
with arbitrary
$\unicode[STIX]{x1D700}>0$
.
For the prime case we need two further tools dealing with the bilinear forms appearing after an application of Vaughan’s identity (Lemma 4.4). These are adoptions of the corresponding Lemmas 3 and 4 of Harman [Reference Harman23] to our setting of pseudo-polynomial functions. Let
$\unicode[STIX]{x1D711}$
and
$\unicode[STIX]{x1D713}$
be two real functions. Then, for
$U$
,
$V$
and
$X$
reals, we consider sums of the form
$$\begin{eqnarray}\mathop{\sum }_{u=1}^{U}\mathop{\sum }_{\substack{ v=1 \\ X<uv\leqslant 2X}}^{V}\unicode[STIX]{x1D711}(u)\unicode[STIX]{x1D713}(v)e(P(uv)+g(uv)),\end{eqnarray}$$
where
$P$
is a polynomial of degree
$k$
and
$g$
is a
$k$
-times continuously differentiable real function. Furthermore, we define
$\unicode[STIX]{x1D6F9}$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}(n,y_{1},\ldots ,y_{s}) & = & \displaystyle \unicode[STIX]{x1D713}(n)\mathop{\prod }_{i=1}^{s}\unicode[STIX]{x1D713}(n+y_{i})\mathop{\prod }_{1\leqslant i<j\leqslant s}\unicode[STIX]{x1D713}(n+y_{i}+y_{j})\nonumber\\ \displaystyle & & \displaystyle \cdots \mathop{\prod }_{i=1}^{s}\unicode[STIX]{x1D713}\biggl(n+\mathop{\sum }_{j\neq i}y_{i}\biggr)\unicode[STIX]{x1D713}\biggl(n+\mathop{\sum }_{i=1}^{s}y_{i}\biggr).\nonumber\end{eqnarray}$$
The first lemma of Harman [Reference Harman23] is the following.
Lemma 5.8. Let
$P$
be a polynomial of degree
$k$
with real coefficients and let
$\unicode[STIX]{x1D6FC}$
be its leading coefficient. Let
$g(x)$
be a real (
$2k+1$
)-times continuously differentiable function on
$[X,2X]$
such that
$\biggl|g^{(r)}(x)\biggr|\asymp GX^{-r}~(r=1,\ldots ,k)$
. Set
$$\begin{eqnarray}T=\max |\unicode[STIX]{x1D713}(v)|\quad \text{and}\quad F=\biggl(\frac{1}{U}\biggl(\mathop{\sum }_{u\leqslant U}\unicode[STIX]{x1D711}(u)^{2}\biggr)\biggr)^{1/2}.\end{eqnarray}$$
For positive integers
$U$
,
$V$
,
$X$
, write
$$\begin{eqnarray}S=\mathop{\sum }_{u=1}^{U}\mathop{\sum }_{\substack{ v=1 \\ X<uv\leqslant 2X}}^{V}\unicode[STIX]{x1D711}(u)\unicode[STIX]{x1D713}(v)e(P(uv)+g(uv)).\end{eqnarray}$$
Suppose that there exist
$a,b\in \mathbb{Z}$
such that
$$\begin{eqnarray}\biggl|\unicode[STIX]{x1D6FC}-\frac{a}{b}\biggr|\leqslant \frac{1}{b^{2}}.\end{eqnarray}$$
Then we have
$$\begin{eqnarray}\biggl(\frac{S}{TF}\biggr)^{K^{2}}\ll (UV)^{K^{2}+\unicode[STIX]{x1D700}}(V^{-K}+(1+G)(U^{-1}+b^{-1}+(UV)^{-k}b)),\end{eqnarray}$$
where
$K=2^{k-1}$
.
Proof. We may assume that
$T=F=1$
and
$\unicode[STIX]{x1D713}(v)\geqslant 0$
for all
$v$
as well as omit the restriction
$X\leqslant uv\leqslant 2X$
for the moment. Then an application of the Cauchy–Schwarz inequality yields
$$\begin{eqnarray}\displaystyle S^{2} & \ll & \displaystyle U\mathop{\sum }_{v_{1}=1}^{V}\mathop{\sum }_{v_{2}=1}^{V}\unicode[STIX]{x1D713}(v_{1})\unicode[STIX]{x1D713}(v_{2})\mathop{\sum }_{u=1}^{U}e(P(uv_{1})-P(uv_{2})+g(uv_{1})-g(uv_{2}))\nonumber\\ \displaystyle & \ll & \displaystyle US_{1}+E_{1}.\nonumber\end{eqnarray}$$
For positive integers
$s$
, we write for short
$$\begin{eqnarray}S_{s}=\mathop{\sum }_{d_{1}=1}^{V-1}\cdots \mathop{\sum }_{d_{s}=1}^{V-1}\mathop{\sum }_{v}\unicode[STIX]{x1D6F9}(v,d_{1},\ldots ,d_{s})\mathop{\sum }_{u=1}^{U}e(\unicode[STIX]{x1D6E5}_{d_{s},\ldots ,d_{1}}(P+g)(uv)),\end{eqnarray}$$
where the forward difference operator
$\unicode[STIX]{x1D6E5}_{d_{s},\ldots ,d_{1}}$
acts on
$v$
not on
$u$
and the range of summation over
$v$
being
$X<v<v+d_{1}+\cdots +d_{s}\leqslant 2X$
, and
$$\begin{eqnarray}E_{s}=U^{2^{s}}V^{2^{s}-1}.\end{eqnarray}$$
An easy induction shows for
$s=2,\ldots ,k-1$
that
$$\begin{eqnarray}S^{2^{s}}\ll E_{s}+U^{2^{s}-1}V^{2^{s}-s-1}|S_{s}|.\end{eqnarray}$$
Now we look at the innermost sum of
$S_{s}$
. Since (cf. [Reference Schmidt48, Lemma 10B])
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(P)(uv)\nonumber\\ \displaystyle & & \displaystyle \quad =d_{1}\cdots d_{k-}({\textstyle \frac{1}{2}}k!\unicode[STIX]{x1D6FC}u^{k}(2v+d_{1}+\cdots +d_{k-1})+(k-1)!\unicode[STIX]{x1D6FD}u^{k-1})\nonumber\\ \displaystyle & & \displaystyle \quad =u^{k}h(d_{1},\ldots ,d_{k-1},v)+u^{k-1}(k-1)!\unicode[STIX]{x1D6FD}d_{1}\cdots d_{k-1},\nonumber\end{eqnarray}$$
we can see
$\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(P)(uv)$
as a polynomial of degree
$k$
with leading coefficient
$h(d_{1},\ldots ,d_{k-1},v)$
. Furthermore,
$\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(g)(uv)$
is a
$k$
-times differentiable function and we may apply Lemma 5.5. Thus, after
$k-1$
iterations of the Cauchy–Schwarz inequality, we obtain
$$\begin{eqnarray}\displaystyle S^{K^{2}} & \ll & \displaystyle (UV)^{K^{2}}V^{-K}+U^{K^{2}-K}V^{K^{2}-k}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{d_{1}=1}^{V}\cdots \mathop{\sum }_{d_{k-1}=1}^{V}\mathop{\sum }_{v}\biggl|\mathop{\sum }_{u=1}^{U}e(\unicode[STIX]{x1D6E5}_{d_{k-1},\ldots ,d_{1}}(P+g)(uv))\biggr|^{K}\nonumber\\ \displaystyle & \ll & \displaystyle (UV)^{K^{2}}V^{-K}+(UV)^{K^{2}-k+\unicode[STIX]{x1D700}}(1+G)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{d_{1},\ldots ,d_{k-1}}\mathop{\sum }_{v}\mathop{\sum }_{t=1}^{k!U^{k-1}}\min \biggl(U,\frac{1}{\Vert th(d_{1},\ldots ,d_{k-1},v)\Vert }\biggr).\nonumber\end{eqnarray}$$
There are at most
$(UV)^{\unicode[STIX]{x1D700}}$
ways of writing a number
$t\leqslant (k!)^{2}V^{k}U^{k-1}$
as a product of the form
$$\begin{eqnarray}{\textstyle \frac{1}{2}}d_{1}d_{2}\cdots d_{k-1}(k!)(2v+d_{1}+\cdots +d_{k-1})=t.\end{eqnarray}$$
Thus,
$$\begin{eqnarray}S^{K^{2}}\ll (UV)^{K^{2}}V^{-K}+(UV)^{K^{2}-k+\unicode[STIX]{x1D700}}(1+G)\mathop{\sum }_{m=1}^{(k!)^{2}V^{k}U^{k-1}}\min \biggl(U,\frac{1}{\Vert m\unicode[STIX]{x1D6FC}\Vert }\biggr).\end{eqnarray}$$
Finally, we use Lemma 5.4 for the sum of minima to get the desired bound. ◻
As the second part, we adapt Lemma 4 of Harman [Reference Harman23].
Lemma 5.9. Suppose that we have the hypotheses of Lemma 5.8, but either
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(x) & = & \displaystyle 1\quad \text{for all }x\nonumber\\ \displaystyle \text{or }\unicode[STIX]{x1D711}(x) & = & \displaystyle \log x\quad \text{for all }x.\nonumber\end{eqnarray}$$
Then
$$\begin{eqnarray}S\ll (UV)^{1+\unicode[STIX]{x1D700}}V^{(k-1)/K}(1+G)^{1/K}((UV)^{-k}b+U^{-1}+b^{-1})^{1/K}.\end{eqnarray}$$
Proof. By an application of partial summation, we may easily remove the
$\log$
factor. Therefore, without loss of generality, we assume that
$\unicode[STIX]{x1D711}(x)=1$
. Using Hölder’s inequality, we obtain
$$\begin{eqnarray}S^{K}\leqslant V^{K-1}\mathop{\sum }_{v=1}^{V}\biggl|\mathop{\sum }_{u=1}^{U}e(P(uv)+g(uv))\biggr|^{K},\end{eqnarray}$$
where this time the forward difference operator
$\unicode[STIX]{x1D6E5}_{d_{1},\ldots ,d_{k-1}}$
is with respect to
$u$
.
Now an application of Lemma 5.5 for the innermost sum yields
$$\begin{eqnarray}\displaystyle S^{K} & \ll & \displaystyle V^{K-1}\mathop{\sum }_{v=1}^{V}U^{K-k+\unicode[STIX]{x1D700}}(1+G)\mathop{\sum }_{t=1}^{k!U^{k-1}}\min \biggl(U,\frac{1}{\Vert tv^{k}\unicode[STIX]{x1D6FC}\Vert }\biggr)\nonumber\\ \displaystyle & \ll & \displaystyle U^{K-k+\unicode[STIX]{x1D700}}V^{K-1}(1+G)\mathop{\sum }_{w=1}^{k!U^{k-1}V^{k}}\min \biggl(U,\frac{1}{\Vert w\unicode[STIX]{x1D6FC}\Vert }\biggr).\nonumber\end{eqnarray}$$
An application of Lemma 5.4 proves the lemma.◻
Finally, we have to combine the two lemmas as in the proof of Harman.
Proof of Proposition 5.7.
This proof starts along the same lines as the proof of Proposition 4.5. An application of Lemma 4.3 transforms the sum over the primes into the weighted sum
$$\begin{eqnarray}\biggl|\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D709}g(p)+\unicode[STIX]{x1D709}P(p))\biggr|\ll \frac{1}{\log N}\max \biggl|\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D6EC}(n)e(\unicode[STIX]{x1D709}(g(n)+P(n)))\biggr|+N^{1/2}.\end{eqnarray}$$
Then we split the inner sum into
${\leqslant}\log N$
subsums of the form
$$\begin{eqnarray}\biggl|\mathop{\sum }_{X<n\leqslant 2X}\unicode[STIX]{x1D6EC}(n)e(\unicode[STIX]{x1D709}(g(n)+P(n)))\biggr|\end{eqnarray}$$
with
$2X\leqslant N$
and we denote by
$S$
a typical one of them. We may assume that
$X\geqslant N^{1-\unicode[STIX]{x1D70C}}$
.
Applying Vaughan’s identity (Lemma 4.4) with the parameters
$U=\frac{1}{4}X^{1/5}$
,
$V=4X^{1/3}$
and
$Z$
, the unique number in
$\frac{1}{2}+\mathbb{N}$
which is closest to
$\frac{1}{4}X^{2/5}$
yields
$$\begin{eqnarray}S\ll 1+(\log X)S_{1}+(\log X)^{8}S_{2},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle S_{1} & = & \displaystyle \mathop{\sum }_{x<2X/Z}d_{3}(x)\mathop{\sum }_{y>Z,X/x<y<2X/x}e(\unicode[STIX]{x1D709}(g(xy)+P(xy))),\nonumber\\ \displaystyle S_{2} & = & \displaystyle \mathop{\sum }_{X/V<x\leqslant 2X/U}d_{4}(x)\mathop{\sum }_{U<y<V,X/x<y\leqslant 2X/x}b(y)e(\unicode[STIX]{x1D709}(g(xy)+P(xy))).\nonumber\end{eqnarray}$$
We consider these two sums as variants of the following general sum:
$$\begin{eqnarray}S_{3}=\mathop{\sum }_{u\leqslant 2X/V}\mathop{\sum }_{\substack{ v\leqslant V \\ X<uv\leqslant 2X}}\unicode[STIX]{x1D711}(u)\unicode[STIX]{x1D713}(v)e(\unicode[STIX]{x1D709}f(uv)),\end{eqnarray}$$
where
$V\ll X^{1/3}$
or
$V\ll X^{2/3}$
.
Similarly as in Proposition 5.1, we get that
$$\begin{eqnarray}|\unicode[STIX]{x1D709}g^{(\ell )}(n)|\asymp |\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}-\ell }.\end{eqnarray}$$
Furthermore, let
$a,b\in \mathbb{Z}$
be as in Lemma 5.6, i.e.
$$\begin{eqnarray}\biggl|\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FC}-\frac{a}{b}\biggr|\leqslant b^{-2}\quad \text{and}\quad N^{2\unicode[STIX]{x1D70C}}\leqslant b\leqslant N^{k-2\unicode[STIX]{x1D70C}}.\end{eqnarray}$$
Now we apply Lemmas 5.8 and 5.9 whether
$$\begin{eqnarray}V^{K}\geqslant N^{-\unicode[STIX]{x1D70C}}\min (b,N^{1/3},N^{k}b^{-1})\end{eqnarray}$$
holds or not, respectively. Suppose that (5.2) holds. In this case an application of Lemma 5.8 yields
$$\begin{eqnarray}S_{3}^{K^{2}}\ll X^{K^{2}+\unicode[STIX]{x1D700}}(V^{K}+|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}}(X^{-1/3}+b^{-1}+X^{-k}b)).\end{eqnarray}$$
On the contrary, if (5.2) does not hold, then an application of Lemma 5.9 yields
$$\begin{eqnarray}S_{3}\ll X^{1+\unicode[STIX]{x1D700}}V^{(k-1)/K}(1+|\unicode[STIX]{x1D709}|X^{\unicode[STIX]{x1D703}_{r}})^{1/K}(X^{-k}b+X^{-1/3}+b^{-1})^{1/K},\end{eqnarray}$$
where we have used that
$1/K-(k-1)/K^{2}\geqslant 1/K^{2}$
.
Summing over the intervals
$[X,2X]$
using both estimates together with
$\biggl|\unicode[STIX]{x1D709}\biggr|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}$
yields the desired bound.◻
6 The case of a single pseudo-polynomial
In the present section we want to prove Theorems 2.3 and 2.6. The main tool originates from large-sieve estimates due to Montgomery, which provide us with a lower bound if all elements are sufficiently far away from an integer.
Lemma 6.1. Let
$M$
and
$N$
be positive integers. Consider a sequence of real numbers
$x_{1},\ldots ,x_{N}$
and weights
$c_{1},\ldots ,c_{N}\geqslant 0$
. Suppose that
$\Vert x_{j}\Vert \geqslant M^{-1}$
for all
$j=1,\ldots ,N$
. Then there exists
$1\leqslant m\leqslant M$
such that
$$\begin{eqnarray}\biggl|\mathop{\sum }_{n=1}^{N}c_{n}e(mx_{n})\biggr|\geqslant \frac{1}{6M}\mathop{\sum }_{n=1}^{N}c_{n}.\end{eqnarray}$$
Proof. This is a weighted version of [Reference Baker1, Theorem 2.2].◻
The second tool are Vaaler polynomials, which we need to deal with the floor function.
Lemma 6.2 [Reference Vaaler53, Theorem 19].
Let
$I\subset [0,1]$
be an interval and
$\unicode[STIX]{x1D712}_{I}$
its indicator function. Then for each positive integer
$H$
there exist coefficients
$a_{H}(h)$
and
$C_{h}$
for
$-H\leqslant h\leqslant H$
with
$|a_{H}(h)|\leqslant 1$
and
$|C_{h}|\leqslant 1$
such that the trigonometric polynomial
$$\begin{eqnarray}\unicode[STIX]{x1D712}_{I,H}^{\ast }(t)=|I|+\frac{1}{\unicode[STIX]{x1D70B}}\mathop{\sum }_{0<|h|\leqslant H}\frac{a_{H}(h)}{|h|}e(ht)\end{eqnarray}$$
satisfies
$$\begin{eqnarray}|\unicode[STIX]{x1D712}_{I}(t)-\unicode[STIX]{x1D712}_{I,H}^{\ast }(t)|\leqslant \frac{1}{H+1}\mathop{\sum }_{|h|\leqslant H}C_{h}\biggl(1-\frac{|h|}{H+1}\biggr)e(ht).\end{eqnarray}$$
Remark 6.3. The coefficients
$a_{H}(h)$
and
$C_{h}$
are explicitly given in Vaaler’s proof. However, the stated bounds are sufficient for our purposes.
Now we are able to state the proof of Theorem 2.3.
Proof of Theorem 2.3.
Let
$M=\lfloor N^{\unicode[STIX]{x1D702}}\rfloor$
, where
$\unicode[STIX]{x1D702}$
is a sufficiently small exponent. We conduct the proof by supposing that
$$\begin{eqnarray}\min _{1\leqslant n\leqslant N}\Vert \unicode[STIX]{x1D709}\lfloor f(n)\rfloor \Vert \geqslant M^{-1}\end{eqnarray}$$
and deducing a contradiction. Since
$$\begin{eqnarray}M^{-1}\leqslant \Vert \unicode[STIX]{x1D709}\lfloor f(n)\rfloor \Vert \leqslant \Vert \unicode[STIX]{x1D709}\Vert \cdot \lfloor f(n)\rfloor \ll \Vert \unicode[STIX]{x1D709}\Vert ,\end{eqnarray}$$
we get
$|\unicode[STIX]{x1D709}|\gg M^{-1}$
. Furthermore, by Lemma 6.1, there exists
$1\leqslant m\leqslant M$
such that
$$\begin{eqnarray}\biggl|\mathop{\sum }_{n=1}^{N}e(m\unicode[STIX]{x1D709}\lfloor f(n)\rfloor )\biggr|\gg \frac{N}{M}.\end{eqnarray}$$
The aim is to establish an upper bound of this exponential sum contradicting the lower bound for sufficiently small
$\unicode[STIX]{x1D702}$
.
We suppose for the moment that
$$\begin{eqnarray}\Vert m\unicode[STIX]{x1D709}\Vert \geqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}.\end{eqnarray}$$
The idea is to use digital expansion and Vaaler polynomials to get rid of the floor function. Then we are left with an exponential sum and use the tools from above.
Let
$q\geqslant 2$
be an integer, which is chosen later. Then we denote by
$I_{d}$
with
$0\leqslant d<q-1$
the interval of all reals in
$[0,1)$
whose initial
$q$
-adic digit is
$d$
, i.e.
$$\begin{eqnarray}I_{d}:=\biggl[\frac{d}{q},\frac{d+1}{q}\biggr).\end{eqnarray}$$
If
$\{f(n)\}\in I_{d}$
, then there exists
$0\leqslant \unicode[STIX]{x1D717}<1$
such that
$\{f(n)\}=d/q+\unicode[STIX]{x1D717}/q$
. Thus,
$$\begin{eqnarray}e(m\unicode[STIX]{x1D709}\lfloor f(n)\rfloor )=e\biggl(m\unicode[STIX]{x1D709}f(n)-m\unicode[STIX]{x1D709}\frac{d}{q}\biggr)\biggl(1+{\mathcal{O}}\biggl(\frac{m}{q}\biggr)\biggr),\end{eqnarray}$$
yielding
$$\begin{eqnarray}\biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}\lfloor f(n)\rfloor )\biggr|=\mathop{\sum }_{d=0}^{q-1}\biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\unicode[STIX]{x1D712}_{I_{d}}(f(n))\biggr|+{\mathcal{O}}\biggl(\frac{Nm}{q}\biggr).\end{eqnarray}$$
Hence, for a fixed
$d\in \{0,\ldots ,q-1\}$
, we may write
$$\begin{eqnarray}\displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\unicode[STIX]{x1D712}_{I_{d}}(f(n))\biggr| & {\leqslant} & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\unicode[STIX]{x1D712}_{I_{d}}^{\ast }(f(n))\biggr|\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{n\leqslant N}|\unicode[STIX]{x1D712}_{I_{d}}(f(n))-\unicode[STIX]{x1D712}_{I_{d},H}^{\ast }(f(n))|,\nonumber\end{eqnarray}$$
where we used the notation from Lemma 6.2. Using the estimates there, we get for the first part that
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\unicode[STIX]{x1D712}_{I_{d},H}^{\ast }(f(n))\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad =\biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\biggl(\frac{1}{q}+\frac{1}{\unicode[STIX]{x1D70B}}\mathop{\sum }_{1\leqslant |h|\leqslant H}\frac{a_{H}(h)}{|h|}e(hf(n))\biggr)\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{q}\biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\biggr|+\frac{1}{\unicode[STIX]{x1D70B}}\mathop{\sum }_{0<|h|\leqslant H}\frac{1}{|h|}\biggl|\mathop{\sum }_{n\leqslant N}e((m\unicode[STIX]{x1D709}+h)f(n))\biggr|.\end{eqnarray}$$
For the second part, we again use the estimates in Lemma 6.2 and arrive at
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}|\unicode[STIX]{x1D712}_{I_{d}}(f(n))-\unicode[STIX]{x1D712}_{I_{d},H}^{\ast }(f(n))|\leqslant \frac{1}{H+1}\mathop{\sum }_{|h|\leqslant H}\biggl(1-\frac{|h|}{H+1}\biggr)\biggl|\mathop{\sum }_{n\leqslant N}e(hf(n))\biggr|.\end{eqnarray}$$
The different exponential sums occurring in (6.3) and (6.4) are of the form
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n)),\quad \mathop{\sum }_{n\leqslant N}e((m\unicode[STIX]{x1D709}+h)f(n))\quad \text{and}\quad \mathop{\sum }_{n\leqslant N}e(hf(n)),\end{eqnarray}$$
respectively. We write them in the form
$\sum _{n\leqslant N}e(\unicode[STIX]{x1D6FD}f(n))$
for short.
Since
$\Vert m\unicode[STIX]{x1D709}\Vert \geqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
, we get
$\biggl|\unicode[STIX]{x1D6FD}\biggr|\geqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
and we may distinguish the following two cases.
∙ Suppose that we have
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\leqslant \biggl|\unicode[STIX]{x1D6FD}\biggr|\leqslant N^{1/10}$
. Then we write
$f(x)=P(x)+g(x)$
, where
$P$
is a polynomial of degree
$k$
and
$g(x)=\sum _{j=1}^{r}d_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$1<\unicode[STIX]{x1D703}_{1}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\not \in \mathbb{N}$
for
$1\leqslant j\leqslant r$
. Now an application of Proposition 4.2 yields where$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D6FD}f(n))\ll N^{1-1/(8KL-4K)},\end{eqnarray}$$
$K=2^{k}$
and
$L=2^{\lfloor \unicode[STIX]{x1D703}_{r}\rfloor }$
.∙ Now we suppose that
$N^{1-\unicode[STIX]{x1D70C}-\deg f}\leqslant \biggl|\unicode[STIX]{x1D6FD}\biggr|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}$
. Then an application of Proposition 5.1 yields with$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D6FD}f(n))\ll N^{1-\unicode[STIX]{x1D70C}2^{1-k}+\unicode[STIX]{x1D700}}\end{eqnarray}$$
$\unicode[STIX]{x1D700}>0$
.
We set
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\min (\unicode[STIX]{x1D70C}2^{1-k},(8KL-4K)^{-1}).\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}f(n))\unicode[STIX]{x1D712}_{I_{d}}(f(n))\biggr|\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{1-\unicode[STIX]{x1D6FE}}\biggl(\frac{1}{q}+\mathop{\sum }_{0<h\leqslant H}\frac{1}{h}+\frac{1}{H+1}\mathop{\sum }_{h\leqslant H}\biggl(1-\frac{h}{H+1}\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{1-\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D700}},\nonumber\end{eqnarray}$$
where we have chosen
$H=N^{\unicode[STIX]{x1D6FE}}$
. Finally, we set
$q=\lfloor mN^{\unicode[STIX]{x1D6FE}/2}\rfloor$
. Together with the lower bound from (6.2), we obtain that
$$\begin{eqnarray}N^{1-\unicode[STIX]{x1D702}}\ll \biggl|\mathop{\sum }_{n\leqslant N}e(m\unicode[STIX]{x1D709}\lfloor f(n)\rfloor )\biggr|\ll N^{1-\unicode[STIX]{x1D6FE}/2+\unicode[STIX]{x1D702}+\unicode[STIX]{x1D700}}+N^{1-\unicode[STIX]{x1D6FE}/2},\end{eqnarray}$$
yielding a contradiction for
$4\unicode[STIX]{x1D702}<\unicode[STIX]{x1D6FE}$
.
Now we turn our attention to the case of
$\Vert m\unicode[STIX]{x1D709}\Vert \leqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
. Then there is some
$h\in \mathbb{Z}$
such that
$$\begin{eqnarray}|\unicode[STIX]{x1D6FD}|=|m\unicode[STIX]{x1D709}+h|\leqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}\end{eqnarray}$$
and the coefficient in the second sum of (6.5) might be arbitrarily small, destroying our argument. We note that
$m\neq 1$
since otherwise
$\Vert \unicode[STIX]{x1D709}\Vert \leqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
, contradicting (6.1).
For
$m>1$
, we show that there must be a multiple of
$m$
in the sequence
$(\lfloor f(n)\rfloor )_{n}$
, yielding a contradiction to (6.1). In particular, we set
$X=N^{\unicode[STIX]{x1D6FE}/(1-\unicode[STIX]{x1D70C})}$
, where
$\unicode[STIX]{x1D6FE}$
is as in (6.6) and show that there exists
$1\leqslant n\leqslant X$
such that
$\lfloor f(n)\rfloor$
is a multiple of
$m$
.
Let
$$\begin{eqnarray}f(n)=a_{\ell }m^{\ell }+\cdots +a_{1}m+a_{0}+a_{-1}m^{-1}+\cdots\end{eqnarray}$$
be the
$m$
-adic expansion of
$f(n)$
. Then
$\lfloor f(n)\rfloor$
is a multiple of
$m$
if and only if
$a_{0}=0$
and this is the case if and only if
$m^{-1}f(n)\in [0,1/m)$
.
Let
${\mathcal{N}}(f,X)$
be the number of
$1\leqslant n\leqslant X$
such that
$\lfloor f(n)\rfloor$
is a multiple of
$m$
. Furthermore, let
$\unicode[STIX]{x1D712}$
be the indicator function of the interval
$[0,1/m)$
. Then an application of Vaaler polynomials (Lemma 6.2) yields
$$\begin{eqnarray}\displaystyle \biggl|{\mathcal{N}}(f,X)-\frac{X}{m}\biggr| & {\leqslant} & \displaystyle \mathop{\sum }_{1\leqslant n\leqslant X}\biggl|\unicode[STIX]{x1D712}\biggl(\frac{f(n)}{m}\biggr)-\unicode[STIX]{x1D712}^{\ast }\biggl(\frac{f(n)}{m}\biggr)\biggr|\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{1\leqslant n\leqslant X}\biggl|\unicode[STIX]{x1D712}^{\ast }\biggl(\frac{f(n)}{m}\biggr)-\frac{1}{m}\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{H+1}\mathop{\sum }_{|h|\leqslant H}C_{h}\biggl(1-\frac{|h|}{H+1}\biggr)\mathop{\sum }_{1\leqslant n\leqslant X}e\biggl(h\frac{f(n)}{m}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{\unicode[STIX]{x1D70B}}\mathop{\sum }_{0<|h|\leqslant H}\frac{a_{H}(h)}{|h|}\mathop{\sum }_{1\leqslant n\leqslant X}e\biggl(h\frac{f(n)}{m}\biggr).\nonumber\end{eqnarray}$$
Setting
$H=X^{1/10}$
, we note that
$$\begin{eqnarray}X^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\leqslant X^{\unicode[STIX]{x1D70C}-1}=N^{-\unicode[STIX]{x1D6FE}}\leqslant \frac{h}{m}\leqslant X^{1/10}.\end{eqnarray}$$
Then an application of Proposition 4.2 yields
$$\begin{eqnarray}\biggl|{\mathcal{N}}(f,X)-\frac{X}{m}\biggr|\ll X^{1-1/(8KL-4K)+\unicode[STIX]{x1D700}}.\end{eqnarray}$$
Thus, for sufficiently large
$X$
(which is growing with
$N$
), we get that
${\mathcal{N}}(f,X)\geqslant 1$
and therefore there is a
$1\leqslant n\leqslant X$
such that
$\lfloor f(n)\rfloor =m\cdot r$
with
$r\in \mathbb{Z}$
. Thus,
$$\begin{eqnarray}N^{-\unicode[STIX]{x1D702}}\leqslant \frac{1}{M}\leqslant \Vert \unicode[STIX]{x1D709}\lfloor f(n)\rfloor \Vert \leqslant \Vert m\unicode[STIX]{x1D709}\Vert \cdot |r|\leqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}N^{\unicode[STIX]{x1D6FE}\deg f/(1-\unicode[STIX]{x1D70C})},\end{eqnarray}$$
yielding a contradiction as long as
$$\begin{eqnarray}\unicode[STIX]{x1D702}<\frac{\deg f(\deg f+\unicode[STIX]{x1D70C}-1)}{1-\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D6FE}.\end{eqnarray}$$
Putting both cases together, we get a contradiction if
$$\begin{eqnarray}\unicode[STIX]{x1D702}<\min \biggl(\frac{1}{4},\frac{\deg f(\deg f+\unicode[STIX]{x1D70C}-1)}{1-\unicode[STIX]{x1D70C}}\biggr)\min \biggl(\unicode[STIX]{x1D70C}2^{1-k},\frac{1}{8KL-4K}\biggr),\end{eqnarray}$$
proving the theorem.◻
Proof of Theorem 2.6.
This runs very much along the same lines as the proof of Theorem 2.3 above. Let
$M=\lfloor N^{\unicode[STIX]{x1D702}}\rfloor$
for a sufficiently small
$\unicode[STIX]{x1D702}$
, which we choose later. Suppose that
$\Vert \unicode[STIX]{x1D709}\lfloor f(p)\rfloor \Vert \geqslant M^{-1}$
for all primes
$2\leqslant p\leqslant N$
. An application of Lemma 6.1 yields
$$\begin{eqnarray}\biggl|\mathop{\sum }_{p\leqslant N}e(m\unicode[STIX]{x1D709}\lfloor f(p)\rfloor )\biggr|\gg \frac{\unicode[STIX]{x1D70B}(N)}{M},\end{eqnarray}$$
where
$\unicode[STIX]{x1D70B}$
is the prime-counting function. As in the integer case, we are looking for an upper bound for the exponential sum yielding conditions on
$\unicode[STIX]{x1D702}$
.
We start with the case of
$\Vert m\unicode[STIX]{x1D709}\Vert \geqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
. Following the lines of the integer case, we have to find upper bounds for exponential sums of the form
$$\begin{eqnarray}\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D6FD}f(p))\end{eqnarray}$$
with
$\unicode[STIX]{x1D6FD}=m\unicode[STIX]{x1D709}$
,
$\unicode[STIX]{x1D6FD}=m\unicode[STIX]{x1D709}+h$
and
$\unicode[STIX]{x1D6FD}=h$
, respectively. We again distinguish two cases.
∙ Either we have
$N^{1-\unicode[STIX]{x1D70C}-\deg f}\leqslant \biggl|\unicode[STIX]{x1D6FD}\biggr|\leqslant N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}$
. Then an application of Proposition 5.7 yields $$\begin{eqnarray}\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D6FD}f(p))\ll N^{1-\unicode[STIX]{x1D70C}4^{1-k}+\unicode[STIX]{x1D700}}.\end{eqnarray}$$
∙ Or we have
$N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D703}_{r}}\leqslant \biggl|\unicode[STIX]{x1D6FD}\biggr|\leqslant N^{1/10}$
. Again we write
$f(x)=P(x)+g(x)$
, where
$P$
is a polynomial of degree
$k$
and
$g(x)=\sum _{j=1}^{r}d_{j}x^{\unicode[STIX]{x1D703}_{j}}$
with
$1<\unicode[STIX]{x1D703}_{1}<\cdots <\unicode[STIX]{x1D703}_{r}$
and
$\unicode[STIX]{x1D703}_{j}\not \in \mathbb{N}$
for
$1\leqslant j\leqslant r$
. Then an application of Proposition 4.5 yields where$$\begin{eqnarray}\mathop{\sum }_{p\leqslant N}e(\unicode[STIX]{x1D6FD}f(p))\ll N^{1-1/(64KL^{5}-4K)+\unicode[STIX]{x1D700}},\end{eqnarray}$$
$K=2^{k}$
and
$L=2^{\lfloor \unicode[STIX]{x1D703}_{r}\rfloor }$
.
This time we set
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\min \biggl(\unicode[STIX]{x1D70C}4^{1-k},\frac{1}{64KL^{5}-4K}\biggr).\end{eqnarray}$$
Following the lines above, we arrive at
$$\begin{eqnarray}N^{1-\unicode[STIX]{x1D702}}\ll \biggl|\mathop{\sum }_{p\leqslant N}e(m\unicode[STIX]{x1D709}\lfloor f(p)\rfloor )\biggr|\ll N^{1-\unicode[STIX]{x1D6FE}/2+\unicode[STIX]{x1D702}+\unicode[STIX]{x1D700}}+N^{1-\unicode[STIX]{x1D6FE}/2},\end{eqnarray}$$
yielding a contradiction for
$4\unicode[STIX]{x1D702}<\unicode[STIX]{x1D6FE}$
.
Now we turn our attention to the case of
$\Vert m\unicode[STIX]{x1D709}\Vert \leqslant N^{1-\unicode[STIX]{x1D70C}-\deg f}$
. Again following the integer case above together with Proposition 4.5, we get the existence of a prime
$2\leqslant p\leqslant N^{\unicode[STIX]{x1D6FE}/(1-\unicode[STIX]{x1D70C})}$
, where this time we have
$\unicode[STIX]{x1D6FE}$
as in (6.8) such that
$\lfloor f(p)\rfloor$
is a multiple of
$m$
. Repeating the steps from above, we get a contradiction if
$$\begin{eqnarray}\unicode[STIX]{x1D702}<\frac{\deg f(\deg f+\unicode[STIX]{x1D70C}-1)}{1-\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D6FE}.\end{eqnarray}$$
Together with the first case, this yields a contradiction provided that
$$\begin{eqnarray}\unicode[STIX]{x1D702}<\min \biggl(\frac{1}{4},\frac{\deg f(\deg f+\unicode[STIX]{x1D70C}-1)}{1-\unicode[STIX]{x1D70C}}\biggr)\min \biggl(\unicode[STIX]{x1D70C}4^{1-k},\frac{1}{64KL^{5}-4K}\biggr),\end{eqnarray}$$
proving the theorem.◻
7 The multidimensional case
In this section we turn our attention to the case of simultaneous approximation. The equivalent to Theorem 2.8 is the following more general result.
Theorem 7.1. Let
$f_{1},\ldots ,f_{k}$
be pseudo-polynomials of pairwise different degree
$\deg f_{i}$
,
$\ell \in \mathbb{N}$
and
$N\in \mathbb{N}$
sufficiently large. Then there exists
$\unicode[STIX]{x1D703}_{\ell }>0$
such that for any lattice
$\unicode[STIX]{x1D6EC}$
with determinant
$|\text{det}(\unicode[STIX]{x1D6EC})|\leqslant N^{\unicode[STIX]{x1D703}_{\ell }}$
and any
$\ell \times k$
matrix
$A$
, there exists
$n\in \mathbb{N}$
with
$1\leqslant n\leqslant N$
such that
$$\begin{eqnarray}A\left(\begin{array}{@{}c@{}}\lfloor f_{1}(n)\rfloor \\ \vdots \\ \lfloor f_{k}(n)\rfloor \end{array}\right)\in \unicode[STIX]{x1D6EC}+B_{\ell },\end{eqnarray}$$
where
$B_{\ell }$
is the Euclidean unit ball in
$\mathbb{R}^{\ell }$
.
The role of the lower bound similar to Lemma 6.1 is played by the following multidimensional counterpart.
Lemma 7.2 [Reference Schmidt48, Theorem 14A].
Suppose that
$\unicode[STIX]{x1D6EC}$
is a lattice of full rank in
$\mathbb{R}^{\ell }$
such that
$\unicode[STIX]{x1D6EC}\cap B_{\ell }=\{\mathbf{0}\}$
, where
$B_{\ell }$
denotes the Euclidean unit ball in
$\mathbb{R}^{\ell }$
. Suppose that
$\mathbf{x}_{1},\ldots ,\mathbf{x}_{N}\in \mathbb{R}^{\ell }$
are not in
$\unicode[STIX]{x1D6EC}+B_{\ell }$
. Let
$\unicode[STIX]{x1D700}>0$
and
$$\begin{eqnarray}S_{\mathbf{p}}=\mathop{\sum }_{n=1}^{N}e(\mathbf{x}_{n}\cdot \mathbf{p}).\end{eqnarray}$$
Then, provided
$N$
is sufficiently large in terms of
$\unicode[STIX]{x1D700}$
, there are a point
$\mathbf{p}$
in a basis of the dual lattice
$\unicode[STIX]{x1D6F1}$
of
$\unicode[STIX]{x1D6EC}$
such that
$\biggl|\mathbf{p}\biggr|\leqslant N^{\unicode[STIX]{x1D700}}$
and an integer
$1\leqslant t\leqslant N^{\unicode[STIX]{x1D700}}/\biggl|\mathbf{p}\biggr|$
such that
$$\begin{eqnarray}|S_{t\mathbf{p}}|\geqslant N^{1-\unicode[STIX]{x1D700}}|\text{det}(\unicode[STIX]{x1D6EC})|^{-1}.\end{eqnarray}$$
Proof of Theorem 7.1.
As in the one-dimensional case, we use Lemma 7.2 to transform the problem into an estimation of an exponential sum. In particular, suppose that
$\mathbf{x}_{1},\ldots ,\mathbf{x}_{N}$
are not in
$\unicode[STIX]{x1D6EC}+B_{\ell }$
. Then by Lemma 7.2 there exist
$\mathbf{p}$
with
$\biggl|\mathbf{p}\biggr|\leqslant N^{\unicode[STIX]{x1D700}}$
and an integer
$1\leqslant m\leqslant N^{\unicode[STIX]{x1D700}}/\biggl|\mathbf{p}\biggr|$
such that
$$\begin{eqnarray}\mathop{\sum }_{n=1}^{N}e(mp_{1}\unicode[STIX]{x1D709}_{1}\lfloor f_{1}(n)\rfloor +\cdots +mp_{k}\unicode[STIX]{x1D709}_{k}\lfloor f_{k}\rfloor )\geqslant N^{1-\unicode[STIX]{x1D700}}\det (\unicode[STIX]{x1D6EC})^{-1}.\end{eqnarray}$$
Similarly to the above, we derive an estimate for the exponential sum contradicting this lower bound.
We start by defining the necessary parameters
$\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{k}$
which play the role of
$\unicode[STIX]{x1D6FE}$
in the multidimensional case. By reordering if necessary, we may suppose that
$\deg f_{1}<\deg f_{2}<\cdots <\deg f_{k}$
. We split each
$f_{i}(n)=g_{i}(n)+P_{i}(n)$
, where
$P_{i}$
is a polynomial of degree
$k_{i}$
and
$g_{i}(n)=\sum _{j=1}^{r}\unicode[STIX]{x1D6FC}_{j}x^{\unicode[STIX]{x1D703}_{i,j}}$
with
$1<\unicode[STIX]{x1D703}_{i,1}<\cdots <\unicode[STIX]{x1D703}_{i,r}$
and
$\unicode[STIX]{x1D703}_{i,j}\not \in \mathbb{Z}$
for
$j=1,\ldots ,r$
.
Let
$\unicode[STIX]{x1D6FE}_{0}=6\unicode[STIX]{x1D70C}$
and let
$\unicode[STIX]{x1D6FD}_{1}$
be real such that
$\Vert \unicode[STIX]{x1D6FD}_{1}\Vert >N^{\unicode[STIX]{x1D6FE}_{0}/2-\deg f_{1}}$
. Then an application of Propositions 4.2 and 5.5, respectively, yields
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\unicode[STIX]{x1D6FD}_{1}f_{1}(n))\ll N^{1-\unicode[STIX]{x1D6FE}_{1}},\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{1}=\min (\unicode[STIX]{x1D70C}2^{1-k_{1}},(8KL-4K)^{-1}).\end{eqnarray}$$
Now we recursively define
$\unicode[STIX]{x1D6FE}_{j}$
for
$2\leqslant j\leqslant k$
as follows. Let
$\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{j}$
be real such that
$\Vert \unicode[STIX]{x1D6FD}_{j}\Vert >N^{\unicode[STIX]{x1D6FE}_{j-1}/2-\deg f_{j}}$
; then, by Propositions 4.2 and 5.5, respectively, there exists
$\unicode[STIX]{x1D6FE}_{j}>0$
such that
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e\biggl(\mathop{\sum }_{i=1}^{j}\unicode[STIX]{x1D6FD}_{i}f_{i}(n)\biggr)\ll N^{1-\unicode[STIX]{x1D6FE}_{j}}.\end{eqnarray}$$
As in the one-dimensional case, we will distinguish whether there is a large coefficient
$\unicode[STIX]{x1D6FD}_{i}$
or they are all small. Starting with the polynomial of largest degree
$f_{k}$
, we look at its leading coefficient
$mp_{k}\unicode[STIX]{x1D709}_{k}$
. If
$\Vert mp_{k}\unicode[STIX]{x1D709}_{k}\Vert >N^{\unicode[STIX]{x1D6FE}_{k-1}/2-\deg f_{k}}$
, then we get rid of the floor function and apply Propositions 4.2 and 5.5, respectively. If
$\Vert mp_{k}\unicode[STIX]{x1D709}_{k}\Vert \leqslant N^{\unicode[STIX]{x1D6FE}_{k-1}/2-\deg f_{k}}$
, then we continue with the polynomial of second largest degree
$f_{k-1}$
and look at its leading coefficient
$mp_{k-1}\unicode[STIX]{x1D709}_{k-1}$
. If
$\Vert mp_{k-1}\unicode[STIX]{x1D709}_{k-1}\Vert >N^{\unicode[STIX]{x1D6FE}_{k-2}/2-\deg f_{k-1}}$
, then we first get rid of the floor function. We use partial summation to split the largest polynomial apart and apply Propositions 4.2 and 5.5, respectively. Otherwise we continue with the polynomial of third largest degree and so on.
For the moment, we suppose that this stops after
$k-s$
steps. Thus, there exists
$1\leqslant s\leqslant k$
such that
$\Vert mp_{j}\unicode[STIX]{x1D709}_{j}\Vert \leqslant N^{\unicode[STIX]{x1D6FE}_{j-1}/2-\deg _{j}}$
for
$s<j\leqslant k$
, but
$\Vert mp_{s}\unicode[STIX]{x1D709}_{s}\Vert >N^{\unicode[STIX]{x1D6FE}_{s-1}/2-\deg f_{s}}$
. We start by getting rid of the floor function. To this end, we set
$$\begin{eqnarray}q:=\lfloor mN^{\unicode[STIX]{x1D6FE}_{s}/4k}\rfloor .\end{eqnarray}$$
Again we denote by
$I_{d}$
the interval of all reals in
$[0,1]$
whose
$q$
-adic expansion starts with
$d$
. Then if
$\{f_{i}(n)\}\in I_{d}$
there exists
$0\leqslant \unicode[STIX]{x1D717}<1$
such that
$\{f_{i}(n)\}=d/q+\unicode[STIX]{x1D717}/q$
. Thus,
$$\begin{eqnarray}e(mp_{i}\unicode[STIX]{x1D709}_{i}\lfloor f_{i}(n)\rfloor )=e\biggl(mp_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)-mp_{i}\unicode[STIX]{x1D709}_{i}\frac{d}{q}\biggr)\biggl(1+{\mathcal{O}}\biggl(\frac{m}{q}\biggr)\biggr).\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}\lfloor f_{i}(n)\rfloor \biggr)\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d_{1}=0}^{q-1}\cdots \mathop{\sum }_{d_{k}=0}^{q-1}\biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)\biggr)\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}}}(f_{j}(n))\biggr|+{\mathcal{O}}\biggl(\frac{Nm}{q}\biggr).\nonumber\end{eqnarray}$$
As above, we want to approximate the occurring product of indicator functions by suitable functions. We follow Grabner [Reference Grabner20] (cf. [Reference Drmota and Tichy18, §1.2.2]), who considered multidimensional variants of Vaaler polynomials. For a fixed vector of digits
$\mathbf{d}=(d_{1},\ldots ,d_{k})$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)\biggr)\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}}}(f_{j}(n))\biggr|\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant \biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)\biggr)\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}},H}^{\ast }(f_{j}(n))\biggr|+R(H),\end{eqnarray}$$
where
$$\begin{eqnarray}R(H)=\mathop{\sum }_{n\leqslant N}\biggl|\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}}}(f_{j}(n))-\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}},H}^{\ast }(f_{j}(n))\biggr|.\end{eqnarray}$$
We start estimating
$R(H)$
. Using the inequality
$$\begin{eqnarray}\biggl|\mathop{\prod }_{j=1}^{k}b_{j}-\mathop{\prod }_{j=1}^{k}a_{j}\biggr|\leqslant \mathop{\sum }_{\emptyset \neq J\subset \{1,\ldots ,k\}}\mathop{\prod }_{j\not \in J}|a_{j}|\mathop{\prod }_{j\in J}|b_{j}-a_{j}|,\end{eqnarray}$$
we get
$$\begin{eqnarray}\displaystyle \biggl|\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{j}}(f_{j}(n))-\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{j},H}^{\ast }(f_{j}(n))\biggr| & {\leqslant} & \displaystyle \mathop{\sum }_{\emptyset \neq J\subset \{1,\ldots ,k\}}\mathop{\prod }_{j\in J}|\unicode[STIX]{x1D712}_{I_{j}}-\unicode[STIX]{x1D712}_{I_{j},H}^{\ast }|\nonumber\\ \displaystyle & = & \displaystyle b\mathop{\prod }_{j=1}^{k}(1+|\unicode[STIX]{x1D712}_{I_{j}}-\unicode[STIX]{x1D712}_{I_{j},H}^{\ast }|)-1.\nonumber\end{eqnarray}$$
Inserting this into (7.3) together with Lemma 6.2 yields
$$\begin{eqnarray}\displaystyle R(H) & = & \displaystyle \mathop{\sum }_{n\leqslant N}\biggl|\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}}}(f_{j}(n))-\mathop{\prod }_{j=1}^{k}\unicode[STIX]{x1D712}_{I_{d_{j}},H}^{\ast }(f_{j}(n))\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{n\leqslant N}\biggl(\mathop{\prod }_{j=1}^{k}(1+|\unicode[STIX]{x1D712}_{I_{j}}-\unicode[STIX]{x1D712}_{I_{j},H}^{\ast }|)-1\biggr)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{n\leqslant N}\biggl(\mathop{\prod }_{j=1}^{k}\biggl(1+\frac{1}{H+1}+\frac{1}{H+1}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{1\leqslant |h_{j}|\leqslant H}C_{h_{j}}\biggl(1-\frac{|h_{j}|}{H+1}\biggr)e(h_{j}f_{j}(n))\biggr)-1\biggr)\nonumber\\ \displaystyle & = & \displaystyle N\biggl(\biggl(1+\frac{1}{H+1}\biggr)^{k}-1\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{0<\Vert \mathbf{h}\Vert _{\infty }\leqslant H}\biggl(\frac{1}{H+1}\biggr)^{k-\unicode[STIX]{x1D6FF}(\mathbf{h})}\biggl(1+\frac{1}{H+1}\biggr)^{\unicode[STIX]{x1D6FF}(\mathbf{h})}\biggl|\mathop{\sum }_{n\leqslant N}e(\mathbf{h}\cdot \mathbf{f}(n))\biggr|,\nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FF}(\mathbf{h})=\sum _{j=1}^{k}\unicode[STIX]{x1D6FF}_{h_{j}0}$
counts the number of coordinates of
$\mathbf{h}=(h_{1},\ldots ,h_{k})$
which are zero;
$\Vert \mathbf{h}\Vert _{\infty }=\max \{|h_{1}|,\ldots ,|h_{k}|\}$
.
Now we turn our attention to the first part of (7.2). Again using Lemma 6.2, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)\biggr)\mathop{\prod }_{j=1}^{s}\unicode[STIX]{x1D712}_{I_{d_{j}},H}^{\ast }(f_{j}(n))\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad =\biggl|\mathop{\sum }_{n\leqslant N}e\biggl(m\mathop{\sum }_{i=1}^{k}p_{i}\unicode[STIX]{x1D709}_{i}f_{i}(n)\biggr)\mathop{\prod }_{j=1}^{s}\biggl(\frac{1}{q}+\frac{1}{\unicode[STIX]{x1D70B}}\mathop{\sum }_{1\leqslant |h_{j}|\leqslant H}\frac{a_{H}(h_{j})}{|h_{j}|}e(h_{j}f_{j}(n))\biggr)\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{q^{s}}\biggl|\mathop{\sum }_{n\leqslant N}e(m\pmb{\unicode[STIX]{x1D6FC}}\cdot \mathbf{f}(n))\biggr|+\mathop{\sum }_{0<\Vert \mathbf{h}\Vert _{\infty }\leqslant H}\frac{1}{r(\unicode[STIX]{x1D70B}\mathbf{h})}\biggl|\mathop{\sum }_{n\leqslant N}e((m\pmb{\unicode[STIX]{x1D6FC}}+\mathbf{h})\cdot \mathbf{f}(n))\biggr|,\nonumber\end{eqnarray}$$
where
$\pmb{\unicode[STIX]{x1D6FC}}=(p_{1}\unicode[STIX]{x1D709}_{1},\ldots ,p_{k}\unicode[STIX]{x1D709}_{k})$
and
$\mathbf{f}(n)=(f_{1}(n),\ldots ,f_{k}(n))$
.
The occurring exponential sums are of the form
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e(\mathbf{h}\cdot \mathbf{f}(n)),\quad \mathop{\sum }_{n\leqslant N}e(m\pmb{\unicode[STIX]{x1D6FC}}\cdot \mathbf{f}(n))\quad \text{or}\quad \mathop{\sum }_{n\leqslant N}e((m\pmb{\unicode[STIX]{x1D6FC}}+\mathbf{h})\cdot \mathbf{f}(n)).\end{eqnarray}$$
We consider these three sums simultaneously and denote them by
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}e\biggl(\mathop{\sum }_{i=1}^{k}\unicode[STIX]{x1D6FD}_{i}f_{i}(n)\biggr).\end{eqnarray}$$
Using partial summation, we split those
$j>s$
apart. To this end, we set
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}(n)=e\biggl(m\mathop{\sum }_{i=1}^{s}\unicode[STIX]{x1D6FD}_{i}f_{i}(n)\biggr)\quad \text{and}\quad \unicode[STIX]{x1D713}(n)=e\biggl(m\mathop{\sum }_{i=s+1}^{k}\unicode[STIX]{x1D6FD}_{i}f_{i}(n)\biggr). & & \displaystyle \nonumber\end{eqnarray}$$
Since
$\unicode[STIX]{x1D6FE}_{j-1}>\unicode[STIX]{x1D6FE}_{j}$
for
$s<j\leqslant k$
, we get
$$\begin{eqnarray}|\unicode[STIX]{x1D713}(n+1)-\unicode[STIX]{x1D713}(n)|\ll \mathop{\sum }_{i=s+1}^{k}N^{\unicode[STIX]{x1D6FE}_{i-1}/2-\deg f_{i}}N^{\deg f_{i}-1}\ll N^{\unicode[STIX]{x1D6FE}_{s}/2-1}.\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D719}(n)\unicode[STIX]{x1D713}(n)\leqslant \unicode[STIX]{x1D713}(N)\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D719}(n)+N^{\unicode[STIX]{x1D6FE}_{s}/2-1}\mathop{\sum }_{t\leqslant N}\mathop{\sum }_{n\leqslant t}\unicode[STIX]{x1D719}(n).\end{eqnarray}$$
By the definition of
$\unicode[STIX]{x1D6FE}_{s}$
, we have
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D719}(n)=\mathop{\sum }_{n\leqslant N}e\biggl(\mathop{\sum }_{i=1}^{s}\unicode[STIX]{x1D6FD}_{i}f_{i}(n)\biggr)\ll N^{1-\unicode[STIX]{x1D6FE}_{s}}.\end{eqnarray}$$
Putting everything together, we get that
$$\begin{eqnarray}N^{1-\unicode[STIX]{x1D702}}\ll \mathop{\sum }_{n\leqslant N}e\biggl(\mathop{\sum }_{i=1}^{k}mp_{i}\unicode[STIX]{x1D709}_{i}\lfloor f_{i}(n)\rfloor \biggr)\ll N^{1-3\unicode[STIX]{x1D6FE}_{s}/4}+N^{1-\unicode[STIX]{x1D6FE}_{s}/4+\unicode[STIX]{x1D702}}+N^{1-\unicode[STIX]{x1D6FE}_{s}/4},\end{eqnarray}$$
yielding a contradiction for
$8\unicode[STIX]{x1D702}<\unicode[STIX]{x1D6FE}_{s}$
.
Now we return to the case that
$\Vert m\unicode[STIX]{x1D709}_{s}\Vert \leqslant N^{\unicode[STIX]{x1D6FE}_{s}/2-\deg f_{s}}$
for
$1\leqslant s\leqslant k$
. Similarly to the one-dimensional case, we may not apply the indicator function and show that there is a joint multiple of
$m$
directly contradicting that there is no element near a point of the lattice
$\unicode[STIX]{x1D6EC}$
. To this end, let
$$\begin{eqnarray}\displaystyle f_{1}(n) & = & \displaystyle a_{1,\ell }m^{\ell }+\cdots +a_{1,1}m+a_{1,0}+a_{1,-1}m^{-1}+\cdots \,,\nonumber\\ \displaystyle \vdots & & \displaystyle \vdots \nonumber\\ \displaystyle f_{s}(n) & = & \displaystyle a_{s,\ell }m^{\ell }+\cdots +a_{s,1}m+a_{s,0}+a_{s,-1}m^{-1}+\cdots \nonumber\end{eqnarray}$$
be the
$m$
-adic expansions of
$f_{1}(n),\ldots ,f_{s}(n)$
. Then
$m\mid \lfloor f_{i}(n)\rfloor$
for
$i=1,\ldots ,k$
if and only if
$a_{i,0}=0$
for
$i=1,\ldots ,k$
if and only if
$m^{-1}f_{i}(n)\in [0,1/m)$
for
$i=1,\ldots ,k$
.
Let
${\mathcal{N}}(\mathbf{f},X)$
denote the number of
$1\leqslant n\leqslant X$
such that
$\biggl|f_{i}(n)\biggr|$
is a multiple of
$m$
for
$1\leqslant i\leqslant k$
. As above,
$X$
will be a power of
$N$
. If
$\unicode[STIX]{x1D712}$
denotes the indicator function of the interval
$[0,1/m)$
, then an application of Vaaler polynomials 6.2 yields
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|{\mathcal{N}}(\mathbf{f},X)-\frac{X}{m^{k}}\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathop{\sum }_{1\leqslant n\leqslant X}\biggl|\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D712}\biggl(\frac{f_{i}(n)}{m}\biggr)-\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D712}^{\ast }\biggl(\frac{f_{i}(n)}{m}\biggr)\biggr|+\mathop{\sum }_{1\leqslant n\leqslant X}\biggl|\mathop{\prod }_{i=1}^{k}\unicode[STIX]{x1D712}^{\ast }\biggl(\frac{f_{i}(n)}{m}\biggr)-\frac{1}{m^{k}}\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant N\biggl(\biggl(1+\frac{1}{H+1}\biggr)^{k}-1\biggr)+\mathop{\sum }_{0<\Vert \mathbf{h}\Vert _{\infty }\leqslant H}\frac{1}{r(\unicode[STIX]{x1D70B}\mathbf{h})}\biggl|\mathop{\sum }_{n\leqslant X}e\biggl(\frac{1}{m}\mathop{\sum }_{i}h_{i}f_{i}(n)\biggr)\biggr|\nonumber\\ \displaystyle & & \displaystyle \qquad +\mathop{\sum }_{0<\Vert \mathbf{h}\Vert _{\infty }\leqslant H}\biggl(\frac{1}{H+1}\biggr)^{k-\unicode[STIX]{x1D6FF}(\mathbf{h})}\biggl(1+\frac{1}{H+1}\biggr)^{\unicode[STIX]{x1D6FF}(\mathbf{h})}\biggl|\mathop{\sum }_{n\leqslant X}e\biggl(\frac{1}{m}\mathop{\sum }_{i}h_{i}f_{i}(n)\biggr)\biggr|,\nonumber\end{eqnarray}$$
where (as above)
$\unicode[STIX]{x1D6FF}(\mathbf{h})=\#\{1\leqslant i\leqslant k:h_{i}=0\}$
. Setting
$X=N^{\unicode[STIX]{x1D6FE}_{k}/(1-\unicode[STIX]{x1D70C})}$
and
$H=X^{1/10}$
, we get by the same reasoning as in the one-dimensional case that for sufficiently large
$N$
we have
${\mathcal{N}}(\mathbf{f},X)\geqslant 1$
. Thus, similarly to the one-dimensional case, there is a
$1\leqslant n\leqslant N$
such that
$\Vert \unicode[STIX]{x1D709}_{i}\lfloor f_{i}(n)\rfloor \Vert$
is very close to an integer. Multiplying by the primitive vector
$\mathbf{p}$
, we get that this is very close to a point in the lattice
$\unicode[STIX]{x1D6EC}$
, violating the assumption.◻
Acknowledgements
We want to thank Imre Rusza for fruitful discussions on the implications of van der Corput sets on Diophantine approximation mentioned in §3.
Parts of this research work were done when the first author was a visiting professor at the Department of Analysis and Number Theory, Graz University of Technology. The author thanks this institution and the doctoral school “Discrete Mathematics” funded by the Austrian Science Fund (FWF). Furthermore, parts of the present paper were worked out when the authors were at the conference “Normal Numbers: Arithmetic, Computational and Probabilistic Aspects” at the Erwin Schrödinger Institute in Vienna. They thank the institute for its hospitality and facilities.
For the realization of the present paper, the first author received support from the Conseil Régional de Lorraine. The second author acknowledges support of the project F 5510-N26 within the special research area “Quasi-Monte Carlo Methods and Applications” funded by the FWF.
We are grateful to an anonymous referee for his careful reading of the first version of the present paper.
