1 Introduction
For
$k\geqslant 2$
, let
$\unicode[STIX]{x1D70C}_{k}$
denote the supremum of positive numbers
$\unicode[STIX]{x1D708}$
for which
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}p^{k}+\unicode[STIX]{x1D6FD}\Vert <p^{-\unicode[STIX]{x1D708}}\end{eqnarray}$$
has infinitely many solutions in primes
$p$
for every irrational
$\unicode[STIX]{x1D6FC}$
and real
$\unicode[STIX]{x1D6FD}$
. Let
$\unicode[STIX]{x1D70E}_{k}$
denote the supremum of positive numbers
$\unicode[STIX]{x1D708}$
for which
$$\begin{eqnarray}\Vert f_{k}(p)\Vert <p^{-\unicode[STIX]{x1D708}}\end{eqnarray}$$
has infinitely many solutions in primes
$p$
whenever
$f_{k}$
is a polynomial of degree
$k$
with irrational leading coefficient. (See Matomaki [Reference Matomaki14] for the case
$k=1$
, which presents different features from
$k=2,3,\ldots .$
)
To state our main result we define the integer
$J=J(f_{k})$
as follows. For
$$\begin{eqnarray}\displaystyle f_{k}(x) & = & \displaystyle \unicode[STIX]{x1D6FC}x^{k}+\unicode[STIX]{x1D6FD},\nonumber\\ \displaystyle J(f_{k}) & = & \displaystyle 2^{k-1}~(k\leqslant 5),\quad J(f_{k})=k(k-1)\quad (k\geqslant 6).\nonumber\end{eqnarray}$$
For other polynomials of degree
$k$
, let
$$\begin{eqnarray}\displaystyle J(f_{k})=2^{k-1}~(k\leqslant 7),\quad J(f_{k})=2k(k-1)\quad (k\geqslant 8). & & \displaystyle \nonumber\end{eqnarray}$$
Theorem 1. The inequality (1.1) has infinitely many solutions for
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708} & {<} & \displaystyle \frac{2}{13}\quad (k=2),\nonumber\\ \displaystyle \unicode[STIX]{x1D708} & {<} & \displaystyle \frac{1}{10}\quad (k=3),\nonumber\\ \displaystyle \unicode[STIX]{x1D708} & {<} & \displaystyle \frac{0.4079}{J(f_{k})}\quad (k\geqslant 4).\nonumber\end{eqnarray}$$
A few remarks are in order. Our first few results are of the form
$\unicode[STIX]{x1D70E}_{2}\geqslant 2/13$
,
$\unicode[STIX]{x1D70E}_{3}\geqslant 1/10$
,
$\unicode[STIX]{x1D70E}_{4}\geqslant 0.0509875$
. Harman [Reference Harman11] obtained
$\unicode[STIX]{x1D70C}_{2}\geqslant 2/13$
, so we do not have a new result for the polynomials
$\unicode[STIX]{x1D6FC}x^{2}+\unicode[STIX]{x1D6FD}$
. Wong [Reference Wong16] obtained lower bounds for
$\unicode[STIX]{x1D70C}_{3},\ldots ,\unicode[STIX]{x1D70C}_{11}$
with
$\unicode[STIX]{x1D70C}_{3}\geqslant 5/56=0.0892\ldots$
and
$\unicode[STIX]{x1D70C}_{4}\geqslant 1/21=0.0476\ldots .$
In [Reference Harman9], Harman showed that
$\unicode[STIX]{x1D70E}_{k}\geqslant 1/2^{2k-1}$
, so that
$\unicode[STIX]{x1D70E}_{2}\geqslant 1/8$
and
$\unicode[STIX]{x1D70E}_{3}\geqslant 1/32$
. Harman [Reference Harman10] gave improvements for
$\unicode[STIX]{x1D70E}_{4},\unicode[STIX]{x1D70E}_{5},\ldots$
including
$\unicode[STIX]{x1D70E}_{4}\geqslant 4/391=0.0102\ldots .$
Asymptotically, Harman [Reference Harman10] showed that
$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{k}\geqslant \frac{1+o(1)}{12k^{2}\log k}.\end{eqnarray}$$
Our improvements depend on obtaining new “arithmetical information” to use in the Harman sieve [Reference Harman11, Reference Harman12]. This amounts to giving upper bounds for trilinear exponential sums, of the form
$$\begin{eqnarray}\mathop{\sum }_{\ell \leqslant L}~c_{\ell }\sum _{N/2<xy<N}b_{y}e(\ell g(mn))\ll N^{1-\unicode[STIX]{x1D702}},\end{eqnarray}$$
where
$L=N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D700}/3}$
and either
$|a_{x}|\leqslant 1$
and
$|b_{y}|\leqslant 1$
(Type II sums) or
$|a_{x}|\leqslant 1$
and
$b_{y}=1$
identically (Type I sums). The point is to get the estimate over wider ranges than can be found in Baker and Harman [Reference Baker and Harman4], the present “state of the art” for monomials. Here
$g$
is obtained from
$f_{k}$
by replacing its leading coefficient
$\unicode[STIX]{x1D6FC}_{k}$
by
$a/q$
, a convergent to
$\unicode[STIX]{x1D6FC}_{k}$
. Several devices come into play. We give a sharper bound in an auxiliary result on the number of solutions
$y\in (Y,2Y]$
of
$$\begin{eqnarray}\biggl\|\frac{say^{3}}{q}\biggr\|<\frac{1}{Z}\end{eqnarray}$$
for a given integer
$s<q$
by slightly adapting a result of Hooley [Reference Hooley13]. For
$k\geqslant 3$
, we give a relatively simple argument that improves the lower bound on
$Y$
in (1.2) from
$Y\gg L^{2}N^{2\unicode[STIX]{x1D702}}$
(essentially) to
$Y\gg LN^{2\unicode[STIX]{x1D702}}$
, in the “Type II” case. For
$k\geqslant 6$
, we use stronger results on simultaneous approximation to the coefficients of a large Weyl sum [Reference Baker3] than those available to the authors of [Reference Baker and Harman4]; these simultaneous approximation results depend on the work of Bourgain et al [Reference Bourgain, Demeter and Guth6].
We end this section with remarks on notation. We write “
$y\sim Y$
” for “
$Y\leqslant y<2Y$
”. We write
$\langle s_{1},\ldots ,s_{k}\rangle$
and
$[s_{1},\ldots ,s_{k}]$
for greatest common divisor and least common multiple. Let
$e(\unicode[STIX]{x1D703})=e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D703}}$
and
$\Vert x\Vert =\min _{n\in \mathbb{Z}}|x-n|$
. Constants implied by “
$O$
” and “
$\ll$
” depend only on
$k$
,
$\unicode[STIX]{x1D700}$
. We suppose that the positive number
$\unicode[STIX]{x1D700}$
is sufficiently small and let
$\unicode[STIX]{x1D702}=\unicode[STIX]{x1D700}/C_{1}(k)$
, where
$C_{1}(k)$
is a suitable large positive constant. Let
$a/q$
be a convergent (with
$q$
sufficiently large) to the continued fraction of
$\unicode[STIX]{x1D6FC}_{k}$
, where
$f_{k}(x)=\unicode[STIX]{x1D6FC}_{k}x^{k}+\cdots +\unicode[STIX]{x1D6FC}_{1}x+\unicode[STIX]{x1D6FC}_{0}$
, and fix
$N$
with
$$\begin{eqnarray}(L_{1}N^{k})^{1/2}\ll q\ll (L_{1}N^{k})^{1/2},\end{eqnarray}$$
where
$L_{1}$
denotes
$2N^{\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D700}/2}$
with
$\unicode[STIX]{x1D70C}$
defined by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}=\frac{2}{13}\,(k=2),\qquad \unicode[STIX]{x1D70C}=\frac{1}{10}\,(k=3),\qquad \unicode[STIX]{x1D70C}=\frac{0.4079}{J(f_{k})}\quad (k\geqslant 4).\end{eqnarray}$$
Clearly (much as in [Reference Baker and Harman4]) it suffices to prove that there is a positive number of primes in the set
$$\begin{eqnarray}A=\biggl\{\frac{N}{2}<n\leqslant N:\Vert g(n)\Vert <L_{1}^{-1}\biggr\}.\end{eqnarray}$$
Note that our definition of
$L$
gives
$L>N^{\unicode[STIX]{x1D700}/7}L_{1}$
; this “increase” compared to
$L_{1}$
is required at the last stage of Lemma 11 below.
2 Small values of a monomial (mod
$q$
)
Let
$1\leqslant Y<q$
,
$1\leqslant D<q$
,
$Z\geqslant 2$
,
$1\leqslant s<q$
. For later use we need to bound the number of solutions of the inequality
$$\begin{eqnarray}\biggl\|\frac{say^{k}}{q}\biggr\|<\frac{1}{Z}\end{eqnarray}$$
for which
$y\in (Y,2Y]$
and
$\langle y,q\rangle \leqslant D$
. Denote this number by
${\mathcal{N}}_{k}(Y,D,Z,s)$
.
Lemma 1. (i) With the above notation, we have
$$\begin{eqnarray}{\mathcal{N}}_{k}(Y,1,Z,s)\ll q^{1+\unicode[STIX]{x1D702}}Z^{-1}.\end{eqnarray}$$
(ii) Whenever
$sD^{k}<q$
, we have
$$\begin{eqnarray}{\mathcal{N}}_{k}(Y,D,Z,s)\ll q^{1+2\unicode[STIX]{x1D702}}Z^{-1}.\end{eqnarray}$$
Proof. For part (i), see [Reference Baker and Harman4, Lemma 6]. To deduce part (ii), it suffices to show that for
$d\,\mid \,q$
the number of
$y\sim Y$
with (2.1) and
$\langle y,q\rangle =d\leqslant D$
is
$$\begin{eqnarray}\ll q^{1+\unicode[STIX]{x1D702}}Z^{-1}.\end{eqnarray}$$
Write
$y=y_{1}d$
,
$q=q_{1}d$
,
$\langle y_{1},q_{1}\rangle =1$
. Then (2.1) implies
$$\begin{eqnarray}\biggl\|\frac{sd^{k-1}y_{1}^{k}}{q_{1}}\biggr\|<Z^{-1}\end{eqnarray}$$
and
$y_{1}\sim Y/d<q_{1}$
. Since
$sd^{k-1}<q_{1}$
, the desired bound follows from part (i). ◻
Lemma 2. Let
$Y$
,
$Z$
be positive numbers in
$[1,N^{3}]$
. Then
$$\begin{eqnarray}\displaystyle {\mathcal{N}}_{3}(Y,q,Z,s) & \ll & \displaystyle Y^{1/2}+N^{\unicode[STIX]{x1D702}}\Bigg(YZ^{-1/4}+Y\biggl(\frac{\langle s,q\rangle }{q}\biggr)^{1/4}\nonumber\\ \displaystyle & & \displaystyle +\,Y^{1/4}q^{1/4}Z^{-1/4}\Bigg).\nonumber\end{eqnarray}$$
Proof. In the case
$\langle s,q\rangle =1$
, this follows from Hooley [Reference Hooley13, Theorem 1]. For the general case, we rewrite (2.1) as
$$\begin{eqnarray}\biggl\|\frac{s_{1}ay^{3}}{q_{1}}\biggr\|<\frac{1}{Z},\end{eqnarray}$$
where
$d=\langle s,q\rangle$
,
$s=s_{1}d$
,
$q=q_{1}d$
,
$\langle s_{1},q_{1}\rangle =1$
.◻
Lemma 3. (i) Let
$s$
,
$Y$
be positive integers less than
$q$
and let
$Z\geqslant 2$
. Then
$$\begin{eqnarray}{\mathcal{N}}_{2}(Y,1,Z,s)\ll q^{\unicode[STIX]{x1D702}}\biggl(\frac{Y+q^{1/2}}{Z^{1/2}}\biggr).\end{eqnarray}$$
(ii) Let
$D\geqslant 1$
. Whenever
$sD^{2}<q$
, in addition to the above hypotheses, we have
$$\begin{eqnarray}{\mathcal{N}}_{2}(Y,D,Z,s)\ll q^{2\unicode[STIX]{x1D702}}\biggl(\frac{Y+q^{1/2}}{Z^{1/2}}\biggr).\end{eqnarray}$$
Proof. For (i), see [Reference Baker and Harman4, Lemma 9]. We deduce (ii) from (i) by an argument used in proving Lemma 1. ◻
Our next task is to “allow
$s$
to vary” in the counting performed in Lemmas 1–3. For
$S_{0}\geqslant 1$
,
$S_{1}\geqslant 1$
, it is convenient to write
$$\begin{eqnarray}\displaystyle {\mathcal{A}} & & \displaystyle (S_{0},S_{1},d_{0},d_{1})\nonumber\\ \displaystyle & & \displaystyle \qquad =\{\!s=s_{0}s_{1}:\langle s_{0},s_{1}\rangle =1,s_{0}\text{ square-full},S_{0}<s_{0}\leqslant 2S_{0},\nonumber\\ \displaystyle & & \displaystyle \qquad \quad ~\,s_{1}\text{ square-free, }S_{1}<s_{1}\leqslant 2S_{1},d_{0}\,\mid \,s_{0},d_{1}\,\mid \,s_{1}\!\}\nonumber\end{eqnarray}$$
whenever
$d_{0}$
,
$d_{1}$
are positive integers.
Lemma 4. For
$S_{0}\leqslant N$
,
$S_{1}\leqslant N$
,
$$\begin{eqnarray}\#{\mathcal{A}}(S_{0},S_{1},d_{0},d_{1})\ll N^{\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}d_{0}^{-1/2}d_{1}^{-1}.\end{eqnarray}$$
Proof. The number of possible
$s_{1}$
here is
$\ll S_{1}d_{1}^{-1}$
. Any
$s_{0}$
occurring can be written
$$\begin{eqnarray}s_{0}=d_{0}uv,\end{eqnarray}$$
where
$p\,\mid \,u$
implies
$p\,\mid \,d_{0}$
and
$v$
is square-full and relatively prime to
$d_{0}$
.
Obviously,
$v\leqslant 2S_{0}/d_{0}$
, so there are
$O(S_{0}^{1/2}d_{0}^{-1/2})$
possible
$v$
.
It remains to show that
$H$
, the number of different
$u$
that can occur in (2.2), is
$O(N^{\unicode[STIX]{x1D702}})$
.
Now, writing
$p_{1}<\cdots <p_{t}$
for the prime divisors of
$d_{0}$
, we find that
$H$
is at most equal to the number of tuples
$(m_{1},\ldots ,m_{t})$
,
$m_{i}$
a non-negative integer, with
$$\begin{eqnarray}m_{1}\log p_{1}+\cdots +m_{t}\log p_{t}\leqslant \log (2S_{0}).\end{eqnarray}$$
A little thought (replace
$p_{1},\ldots ,p_{t}$
in (2.3) by the first
$t$
primes
$q_{1},\ldots ,q_{t}$
) shows that
$$\begin{eqnarray}H\leqslant \unicode[STIX]{x1D6F9}(2S_{0},q_{t}),\end{eqnarray}$$
in the usual notation for smooth numbers. Since
$q_{t}<(1+\unicode[STIX]{x1D700})t\log t$
if
$t$
is large and
$t<(1+\unicode[STIX]{x1D700})\log N/\text{log}\,N$
, we have
$$\begin{eqnarray}H\leqslant \unicode[STIX]{x1D6F9}(2N,2\log N).\end{eqnarray}$$
An appeal to Theorem 1 of de Bruijn [Reference de Bruijn7] now yields
$$\begin{eqnarray}H\ll N^{\unicode[STIX]{x1D702}}\end{eqnarray}$$
and the lemma follows. ◻
Let
$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{k}(Y,Z,S_{0},S_{1}) & = & \displaystyle \#\Bigg\{y\sim Y:\langle y,q\rangle \leqslant N^{\unicode[STIX]{x1D70C}},\biggl\|\frac{say^{k}}{q}\biggr\|<\frac{1}{Z}\nonumber\\ \displaystyle & & \displaystyle \quad \text{for some }s\in {\mathcal{A}}(S_{0},S_{1},1,1)\Bigg\}.\nonumber\end{eqnarray}$$
Summing over
$s$
in Lemmas 1 and 3, we obtain
$$\begin{eqnarray}{\mathcal{M}}_{k}(Y,Z,S_{0},S_{1})\ll N^{3\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}\,\frac{q}{Z}\end{eqnarray}$$
whenever
$1\leqslant Y<q$
,
$Z\geqslant 2$
, and
$$\begin{eqnarray}4S_{0}S_{1}N^{k\unicode[STIX]{x1D70C}}<q,\end{eqnarray}$$
while, under the same conditions on
$Y$
,
$Z$
,
$S_{0}$
,
$S_{1}$
,
$$\begin{eqnarray}{\mathcal{M}}_{2}(Y,Z,S_{0},S_{1})\ll N^{3\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}(Y+q^{1/2})Z^{-1/2}.\end{eqnarray}$$
To obtain a bound for
${\mathcal{M}}_{3}(Y,Z,S_{0},S_{1})$
, we restrict
$s=s_{0}s_{1}$
to values with
$\langle s_{0},q\rangle =d_{0}$
,
$\langle s_{1},q\rangle =d_{1}$
, at a cost of a factor
$O(N^{\unicode[STIX]{x1D702}})$
. Now Lemmas 2 and 4 together yield
$$\begin{eqnarray}{\mathcal{M}}_{3}(Y,Z,S_{0},S_{1})\ll N^{3\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}(Y^{1/2}+YZ^{-1/4}+Yq^{-1/4}+Y^{1/4}q^{1/4}Z^{-1/4})\end{eqnarray}$$
(the factor
$(d_{0}d_{1})^{1/4}$
in the third term in the bound in Lemma 2 is cancelled by the factor
$d_{0}^{-1/2}d_{1}^{-1}$
in Lemma 4).
In our applications we shall always have (2.5). If we assume this additional condition for
$k=3$
, there are no solutions of
$$\begin{eqnarray}\biggl\|\frac{say^{3}}{q}\biggr\|<\frac{1}{q}\quad \text{with }\langle y,q\rangle \leqslant N^{\unicode[STIX]{x1D70C}}\end{eqnarray}$$
counted in (2.7). So, we may suppose that
$Z<q$
and we obtain
$$\begin{eqnarray}{\mathcal{M}}_{3}(Y,Z,S_{0},S_{1})\ll N^{3\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}(Y^{1/2}+YZ^{-1/4}+Y^{1/4}q^{1/4}Z^{-1/4}).\end{eqnarray}$$
3 Type I sums
Our most basic tool is obtained by combining [Reference Baker1, Theorem 5.1] (with a correction in [Reference Baker2]) and [Reference Baker3, Theorem 4].
Lemma 5. Let
$f$
be a polynomial of degree
$k$
,
$f(x)=\unicode[STIX]{x1D6FE}_{k}x^{k}+\cdots +\unicode[STIX]{x1D6FE}_{1}x+\unicode[STIX]{x1D6FE}_{0}$
. Let
$M\geqslant 1$
and
$X\geqslant 1$
, with
$M=1$
when
$J=J(f)\neq 2^{k-1}$
. Suppose that for some subinterval
$I$
of
$[X/2,X]$
, we have
$$\begin{eqnarray}\mathop{\sum }_{m=1}^{M}\Bigg|\mathop{\sum }_{x\in I}e(mf(x))\Bigg|\geqslant P\geqslant MX^{1-1/J+\unicode[STIX]{x1D702}}.\end{eqnarray}$$
Then there are natural numbers
$s$
and integers
$u_{1},\ldots ,u_{k}$
with
$\langle s,u_{1},\ldots ,u_{k}\rangle =1$
;
$\langle s,u_{2},\ldots ,u_{k}\rangle \leqslant MX^{\unicode[STIX]{x1D702}}$
, in the case
$J(f)=2^{k-1}$
;
$$\begin{eqnarray}\displaystyle & \displaystyle s\ll (MXP^{-1})^{k}X^{\unicode[STIX]{x1D702}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle |s\unicode[STIX]{x1D6FE}_{j}-u_{j}|\ll M^{-1}(MXP^{-1})^{k}X^{\unicode[STIX]{x1D702}-j}\quad (j=1,2,\ldots ,k). & \displaystyle \nonumber\end{eqnarray}$$
Lemma 6. Let
$u\in \mathbb{Z}$
,
$d\in \mathbb{N}$
,
$B\geqslant 1$
,
$L\geqslant 1$
. Let
${\mathcal{N}}$
be the number of solutions of
$$\begin{eqnarray}\ell u\equiv b\hspace{0.6em}({\rm mod}\hspace{0.2em}d)\quad (1\leqslant \ell \leqslant L,1\leqslant b\leqslant B).\end{eqnarray}$$
Then
$$\begin{eqnarray}{\mathcal{N}}\leqslant \min (L,B)+\frac{BL}{d}.\end{eqnarray}$$
Proof. The congruence has no solution unless
$\langle u,d\rangle \,\mid \,b$
. For fixed
$b$
, the number of possibilities for
$\ell (\operatorname{mod}d/\langle u,d\rangle )$
is at most 1. Hence,
$$\begin{eqnarray}{\mathcal{N}}\leqslant \frac{B}{\langle u,d\rangle }\,\biggl(\frac{L\langle u,d\rangle }{d}+1\biggr)\leqslant \frac{BL}{d}+B.\end{eqnarray}$$
On the other hand, for given
$\ell$
, the number of possible
$b$
is at most
$B/d+1$
. This gives the alternative upper bound
$BL/d+L$
.◻
Lemma 7. Let
$k\geqslant 2$
. Let
$f(x)=\unicode[STIX]{x1D6FE}_{k}x^{k}+\cdots +\unicode[STIX]{x1D6FE}_{1}x$
. Let
$X\geqslant 1$
,
$1\leqslant L\leqslant X$
. Suppose that there are integers
$s$
,
$u_{1},\ldots ,u_{k}$
,
$s\leqslant X$
,
$\langle s,u_{1},\ldots ,u_{k}\rangle =1$
and, if
$J(f)=2^{k-1}$
,
$\langle s,u_{2},\ldots ,u_{k}\rangle \ll LN^{\unicode[STIX]{x1D702}}$
such that
$$\begin{eqnarray}|s\unicode[STIX]{x1D6FE}_{j}-u_{j}|\leqslant (2k^{2})^{-1}L^{-1}X^{1-j}\quad (1\leqslant j\leqslant k).\end{eqnarray}$$
Let
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}_{j}=\unicode[STIX]{x1D6FE}_{j}-\frac{u_{j}}{s},\qquad F(x)=\mathop{\sum }_{j=1}^{k}\unicode[STIX]{x1D6FD}_{j}x^{j},\qquad G(x)=\mathop{\sum }_{j=1}^{k}u_{j}x^{j}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle S(s,\ell G)=\mathop{\sum }_{v=1}^{s}e\biggl(\frac{\ell G(v)}{s}\biggr). & \displaystyle \nonumber\end{eqnarray}$$
Then we have, for any subinterval
$I$
of
$[X/2,X]$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{n\in I}e(\ell f(x))-s^{-1}S(s,\ell G)\int _{I}e(\ell F(z))\,dz\Bigg|\nonumber\\ \displaystyle & & \displaystyle \qquad \ll \left\{\begin{array}{@{}ll@{}}N^{2\unicode[STIX]{x1D702}}Ls^{1-1/k}\quad & \text{if }J(f)=2^{k-1},\\ N^{2\unicode[STIX]{x1D702}}(Ls^{1-1/k}+s)\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
Proof. Following the proof of [Reference Baker1, Lemma 4.4], we find that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{n\in I}e(\ell f(x))-s^{-1}S(s,\ell G)\mathop{\sum }_{n\in I}e(\ell F(n))\nonumber\\ \displaystyle & & \displaystyle \qquad \ll s^{-1}\mathop{\sum }_{b=1}^{s-1}\biggl\|\frac{b}{s}\biggr\|^{-1}\Bigg|\mathop{\sum }_{v=1}^{s}e\biggl(\frac{\ell G(v)+bv}{s}\biggr)\Bigg|\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\in I}e(\ell F(n))=\int _{I}e(\ell F(z))\,dz+O(1). & & \displaystyle \nonumber\end{eqnarray}$$
Moreover, by a standard estimate [Reference Cochrane8],
$$\begin{eqnarray}\displaystyle S(s,\ell G) & \ll & \displaystyle \langle \ell ,s\rangle ^{1/k}s^{1-1/k}N^{\unicode[STIX]{x1D702}},\nonumber\\ \displaystyle \mathop{\sum }_{v=1}^{s}e\biggl(\frac{\ell G(v)+bv}{s}\biggr) & \ll & \displaystyle \langle \ell u_{1}+b,\ell u_{2},\ldots ,\ell u_{k},s\rangle ^{1/k}s^{1-1/k}N^{\unicode[STIX]{x1D702}}\nonumber\\ \displaystyle & \ll & \displaystyle D_{\ell }s^{1-1/k}N^{\unicode[STIX]{x1D702}},\nonumber\end{eqnarray}$$
where
$D_{\ell }=\min (L^{2}N^{\unicode[STIX]{x1D702}},\langle \ell u_{1}+b,s\rangle )^{1/k}$
if
$J(f)=1$
and
$D_{\ell }=\langle \ell u_{1}+b,s\rangle$
otherwise.
It follows that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\in I} & & \displaystyle e(\ell f(x))-s^{-1}S(s,\ell G)\int _{I}e(\ell F(z))\,dz\nonumber\\ \displaystyle & & \displaystyle \ll \langle \ell ,s\rangle ^{1/k}s^{1-1/k}N^{\unicode[STIX]{x1D702}}+N^{\unicode[STIX]{x1D702}}s^{-1/k}\mathop{\sum }_{b=1}^{s-1}\biggl\|\frac{b}{s}\biggr\|^{-1}D_{\ell }.\nonumber\end{eqnarray}$$
We now sum the absolute values of the left-hand side over
$\ell \leqslant L$
. Since the contribution from
$\sum _{\ell =1}^{L}\langle \ell ,s\rangle ^{1/k}$
is
$\ll LN^{\unicode[STIX]{x1D702}}$
, a splitting-up argument shows that we need only prove the bound
$$\begin{eqnarray}\ll \left\{\begin{array}{@{}ll@{}}N^{\unicode[STIX]{x1D702}}Ls^{1-1/k}\quad & (J(f)=2^{k-1}),\\ N^{\unicode[STIX]{x1D702}}(Ls^{1-1/k}+s)\quad & \text{(otherwise)}\end{array}\right.\end{eqnarray}$$
for the quantity
$$\begin{eqnarray}\frac{s^{1-1/k}}{B}~\mathop{\sum }_{B/2\leqslant b<2B}~\mathop{\sum }_{\substack{ \ell =1 \\ d\,\mid \,\ell u+b}}^{L}A_{d},\end{eqnarray}$$
where
$d\,\mid \,s$
and
$$\begin{eqnarray}A_{d}=\left\{\begin{array}{@{}ll@{}}\min (L^{2/k}N^{\unicode[STIX]{x1D702}},d^{1/k})\quad & \text{if }J(f)=1,\\ d^{1/k}\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$
Applying Lemma 6, the left-hand side of (3.3) is
$$\begin{eqnarray}\ll \frac{s^{1-1/k}}{B}~\min (L^{2/k}N^{\unicode[STIX]{x1D702}},d^{1/k})\biggl(\frac{BL}{d}+B\biggr)\ll N^{\unicode[STIX]{x1D702}}s^{1-1/k}L\end{eqnarray}$$
if
$J(f)=2^{k-1}$
. Otherwise we obtain
$$\begin{eqnarray}\ll \frac{s^{1-1/k}}{B}~d^{1/k}\biggl(\frac{BL}{d}+B\biggr)\ll s^{1-1/k}L+s.\end{eqnarray}$$
This proves the lemma. ◻
Lemma 8. Let
$k\geqslant 2$
and
$Y\ll N^{1-5\unicode[STIX]{x1D70C}/2}(k=2)$
,
$Y\ll N^{1/2+\unicode[STIX]{x1D70C}}$
$(k\geqslant 3)$
. Then, with
$g$
as defined in §1, we have
$$\begin{eqnarray}\mathop{\sum }_{\ell =1}^{L}~\mathop{\sum }_{y\sim Y}\Bigg|\mathop{\sum }_{n\in I(y)}e(\ell g(yn))\Bigg|\ll N^{1-2\unicode[STIX]{x1D702}},\end{eqnarray}$$
where
$I(y)=(N/2y,N/y]$
.
Proof. Let
${\mathcal{S}}$
be the set of
$y\sim Y$
with
$\langle y,q\rangle \leqslant N^{\unicode[STIX]{x1D70C}}$
and
$$\begin{eqnarray}\mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{n\in I(y)}e(\ell g(yn))\Bigg|>N^{1-2\unicode[STIX]{x1D702}}Y^{-1}.\end{eqnarray}$$
It suffices to show that
$$\begin{eqnarray}T:=\mathop{\sum }_{y\in {\mathcal{S}}}~\mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{n\in I(y)}e(\ell g(yn))\Bigg|\ll N^{1-2\unicode[STIX]{x1D702}}.\end{eqnarray}$$
To see this, the contribution in (3.4) from
$y\sim Y$
for which (3.5) fails is
${<}N^{1-2\unicode[STIX]{x1D702}}$
. The contribution from
$y\sim Y$
for which
$\langle y,q\rangle$
is a fixed divisor
$d$
of
$q$
,
$d>N^{\unicode[STIX]{x1D70C}}$
, is
$$\begin{eqnarray}\ll \frac{Y}{d}\,L\,\frac{N}{Y}\ll N^{1-3\unicode[STIX]{x1D702}}\end{eqnarray}$$
and our claim follows on summing over
$d$
.
Given
$y\in {\mathcal{S}}$
, we apply (3.5) in Lemmas 5 and 7. Here
$\unicode[STIX]{x1D6FE}_{k}=ay^{k}/q$
,
$\unicode[STIX]{x1D6FE}_{j}=\unicode[STIX]{x1D6FC}_{j}y^{j}$
$(j<k)$
. Suppose first that
$J(f)=2^{k-1}$
. Take
$X=N/Y$
,
$M=L$
in Lemma 5. Then
$$\begin{eqnarray}P=N^{1-2\unicode[STIX]{x1D702}}Y^{-1}\geqslant L\biggl(\frac{N}{Y}\biggr)^{1-1/J+\unicode[STIX]{x1D702}}\end{eqnarray}$$
since
$Y\ll N^{1-J\unicode[STIX]{x1D70C}}$
. The integers
$s$
,
$u_{1},\ldots ,u_{k}$
provided by Lemma 5 satisfy (3.1) since
$k\leqslant J$
and so
$$\begin{eqnarray}L^{k}N^{\unicode[STIX]{x1D702}}\ll \frac{N}{Y}\,N^{-\unicode[STIX]{x1D702}}.\end{eqnarray}$$
Lemma 7 yields
$$\begin{eqnarray}\displaystyle \frac{N^{1-2\unicode[STIX]{x1D702}}}{Y} & \ll & \displaystyle \mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{n\in I(y)}e(\ell g(n))\Bigg|\nonumber\\ \displaystyle & \ll & \displaystyle \mathop{\sum }_{\ell =1}^{L}\biggl|s^{-1}S(s,\ell G)\int _{I(y)}e(\ell F(z))\,dz\biggr|+L^{k}N^{3\unicode[STIX]{x1D702}k},\end{eqnarray}$$
where
$I(y)=(N/2y,N/y]$
. Here we suppress dependence of
$F$
,
$G$
on
$y$
. The last term is of smaller order than
$N^{1-2\unicode[STIX]{x1D702}}/Y$
in (3.6), so that
$$\begin{eqnarray}\frac{N^{1-2\unicode[STIX]{x1D702}}}{Y}\ll \mathop{\sum }_{\ell =1}^{L}\bigg|s^{-1}S(s,\ell G)\int _{I(y)}e(\ell F(z))\,dz\bigg|.\end{eqnarray}$$
We now show that (3.7) also holds when
$J(f)\neq 2^{k-1}$
. Select
$m_{0}=m_{0}(y)$
such that
$$\begin{eqnarray}\Bigg|\mathop{\sum }_{n\in I(y)}e(m_{0}g(n))\Bigg|\geqslant P:=\frac{N^{1-2\unicode[STIX]{x1D702}}}{YL}.\end{eqnarray}$$
We have
$Y\ll N^{1-J\unicode[STIX]{x1D70C}}$
, as is easily verified. Hence,
$$\begin{eqnarray}P\geqslant \biggl(\frac{N}{Y}\biggr)^{1-1/J+\unicode[STIX]{x1D702}}.\end{eqnarray}$$
We apply Lemma 5 with
$f=m_{0}g$
, obtaining integers
$s^{\prime }$
,
$u_{1}^{\prime },\ldots ,u_{k}^{\prime }$
with
$s^{\prime }\ll L^{k}N^{3k\unicode[STIX]{x1D702}}$
,
$$\begin{eqnarray}|s^{\prime }m_{0}\unicode[STIX]{x1D6FE}_{j}-u_{j}^{\prime }|\ll L^{k}N^{3k\unicode[STIX]{x1D702}}\biggl(\frac{N}{Y}\biggr)^{-j}\quad (j=1,\ldots ,k).\end{eqnarray}$$
Let
$d=\langle s^{\prime }m_{0},u_{1}^{\prime },\ldots ,u_{k}^{\prime }\rangle$
and
$s=s^{\prime }m_{0}/d$
,
$u_{j}=u_{j}^{\prime }/d$
. Then
$$\begin{eqnarray}|s\unicode[STIX]{x1D6FE}_{j}-u_{j}|\ll L^{k}N^{3\unicode[STIX]{x1D70C}k}\biggl(\frac{N}{Y}\biggr)^{-j}\ll L^{-1}N^{-\unicode[STIX]{x1D702}}\biggl(\frac{N}{Y}\biggr)^{-(j-1)}\end{eqnarray}$$
since
$J\geqslant k+1$
. Thus, we can apply Lemma 7. In the analogue of (3.6), the second term on the right-hand side is now
$$\begin{eqnarray}\ll (Ls^{1-1/k}+s)N^{\unicode[STIX]{x1D702}}\ll N^{(k+1)\unicode[STIX]{x1D70C}}\ll \frac{N}{Y}\,N^{-3\unicode[STIX]{x1D702}}\end{eqnarray}$$
and we again end up with (3.7).
Factorizing
$s$
as
$s=s_{0}s_{1}$
with
$s_{0}$
square-full,
$s_{1}$
square-free, and
$\langle s_{0},s_{1}\rangle =1$
, we have
$$\begin{eqnarray}s^{-1}S(s,\ell G)\ll \biggl(\frac{s_{0}}{\langle s_{0},\ell \rangle }\biggr)^{-1/k}\biggl(\frac{s_{1}}{\langle s_{1},\ell \rangle }\biggr)^{-1/2}.\end{eqnarray}$$
See Cochrane [Reference Cochrane8] for more general results. The estimate
$$\begin{eqnarray}\int _{I(y)}e(\ell F(z))\,dz\ll \min \biggl(\frac{N}{Y},\ell ^{-1/k}\bigg|\frac{y^{k}a}{q}-\frac{u_{1}}{s}\bigg|^{-1/k}\biggr)\end{eqnarray}$$
is a consequence of Vaughan [Reference Vaughan15, Theorem 7.3]. Putting the trivial estimate in (3.9) together with (3.7) and (3.8), we have
$$\begin{eqnarray}\frac{N^{1-2\unicode[STIX]{x1D702}}}{Y}\ll \mathop{\sum }_{\ell =1}^{L}\langle s,\ell \rangle ^{1/2}\cdot \frac{1}{s_{0}^{1/k}s_{1}^{1/2}}~\frac{N}{Y},\end{eqnarray}$$
whence
$$\begin{eqnarray}s_{0}^{1/k}s_{1}^{1/2}\ll LN^{3\unicode[STIX]{x1D702}}.\end{eqnarray}$$
We now subdivide
${\mathcal{S}}$
into
$O((\log N)^{3})$
classes according to the values of
$s_{0}=s_{0}(y)$
,
$s_{1}=s_{1}(y)$
, and
$$\begin{eqnarray}\bigg|\frac{y^{k}a}{q}-\frac{u_{k}}{s_{0}s_{1}}\bigg|.\end{eqnarray}$$
In each class
${\mathcal{S}}(Z,S_{0},S_{1})$
, we have
$s_{0}\sim S_{0}$
,
$s_{1}\sim S_{1}$
with
$$\begin{eqnarray}S_{0}^{1/k}S_{1}^{1/2}\ll LN^{3\unicode[STIX]{x1D702}}\end{eqnarray}$$
and, with
$Z_{0}$
defined below,
$Z=2^{-j}Z_{0}\geqslant 2$
, also
$$\begin{eqnarray}\frac{1}{2s_{0}s_{1}Z}\leqslant \bigg|\frac{y^{k}a}{q}-\frac{u_{k}}{s_{0}s_{1}}\bigg|<\frac{1}{s_{0}s_{1}Z}\quad \text{or}\quad (\text{if }Z=Z_{0})~\bigg|\frac{y^{k}a}{q}-\frac{u_{k}}{s_{0}s_{1}}\bigg|<\frac{1}{s_{0}s_{1}Z}.\end{eqnarray}$$
Here
$$\begin{eqnarray}L^{-1/k}(Z_{0}S_{0}S_{1})^{1/k}=\frac{N}{Y}.\end{eqnarray}$$
From (3.7)–(3.9), (3.11) and (3.12), there is a class
${\mathcal{S}}^{\ast }={\mathcal{S}}(Z,S_{0},S_{1})$
and an
$L_{0}\in [1,L)$
such that
$$\begin{eqnarray}\displaystyle T & \ll & \displaystyle N^{\unicode[STIX]{x1D702}}\mathop{\sum }_{y\in {\mathcal{S}}^{\ast }}~\mathop{\sum }_{\ell \sim L_{0}}\langle s_{0}(y)s_{1}(y),\ell \rangle ^{1/2}L_{0}^{-1/k}Z^{1/k}S_{1}^{-1/2+1/k}\nonumber\\ \displaystyle & \ll & \displaystyle S_{1}^{-1/2+1/k}L^{1-1/k}N^{2\unicode[STIX]{x1D702}}Z^{1/k}\#~{\mathcal{S}}^{\ast }.\nonumber\end{eqnarray}$$
We can estimate
$\#\,{\mathcal{S}}^{\ast }$
using the results of §2 since, for every
$y\in {\mathcal{S}}^{\ast }$
, there is an
$s\in {\mathcal{A}}(S_{0},S_{1},1,1)$
with
$$\begin{eqnarray}\biggl\|\frac{say^{k}}{q}\biggr\|<\frac{1}{Z}.\end{eqnarray}$$
Thus, in the notation of §2,
$$\begin{eqnarray}T\ll S_{1}^{-1/2+1/k}L^{1-1/k}N^{2\unicode[STIX]{x1D702}}Z^{1/k}{\mathcal{M}}_{k}(Y,Z,S_{0},S_{1}).\end{eqnarray}$$
We now conclude the proof by considering separately the cases
$k=2$
,
$k=3$
, and
$k\geqslant 4$
. It is easy to verify the condition (2.5) needed for our bounds on
${\mathcal{M}}_{k}$
since
$$\begin{eqnarray}S_{0}S_{1}N^{k\unicode[STIX]{x1D70C}}\ll N^{2k\unicode[STIX]{x1D70C}}\ll N\ll qN^{-\unicode[STIX]{x1D70C}/2}.\end{eqnarray}$$
$\mathbf{k = 2}$
. Recalling (2.6), we deduce from (3.13) that
$$\begin{eqnarray}\displaystyle T & \ll & \displaystyle L^{1/2}N^{5\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}(Y+q^{1/2})\nonumber\\ \displaystyle & \ll & \displaystyle L^{1/2}N^{5\unicode[STIX]{x1D702}}S_{0}^{1/2}S_{1}N^{1-5\unicode[STIX]{x1D70C}/2}\nonumber\end{eqnarray}$$
since
$1/2+\unicode[STIX]{x1D70C}/4<1-5\unicode[STIX]{x1D70C}/2$
. Using (3.11),
$$\begin{eqnarray}T\ll L^{5/2}N^{1+11\unicode[STIX]{x1D702}-5\unicode[STIX]{x1D70C}/2}\ll N^{1-2\unicode[STIX]{x1D702}}.\end{eqnarray}$$
$\mathbf{k = 3}$
. Recalling (2.4) and (2.8),
$$\begin{eqnarray}\displaystyle T & \ll & \displaystyle L^{2/3}N^{4\unicode[STIX]{x1D702}}Z^{1/3}S_{0}^{1/2}S_{1}^{5/6}\min \biggl(\frac{q}{Z},\,Y^{1/2}+\frac{(Y+Y^{1/4}q^{1/4})}{Z^{1/4}}\biggr)\nonumber\\ \displaystyle & \ll & \displaystyle L^{7/3}N^{9\unicode[STIX]{x1D702}}Z^{1/3}\min \biggl(\frac{q}{Z},\,N^{1/4+\unicode[STIX]{x1D70C}/2}+\frac{N^{1/2+\unicode[STIX]{x1D70C}}}{Z^{1/4}}\biggr)\end{eqnarray}$$
since
$Y^{1/4}q^{1/4}\ll N^{1/8+\unicode[STIX]{x1D70C}/4+3/8+\unicode[STIX]{x1D70C}/8}$
. Next,
$$\begin{eqnarray}\displaystyle & & \displaystyle L^{7/3}N^{9\unicode[STIX]{x1D702}}Z^{1/3}\min \biggl(\frac{q}{Z},\,N^{1/4+\unicode[STIX]{x1D70C}/2}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant L^{7/3}N^{9\unicode[STIX]{x1D702}}q^{1/3}N^{(1/4+\unicode[STIX]{x1D70C}/2)2/3}\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{2/3+\unicode[STIX]{x1D70C}(7/3+1/6+1/3)}\ll N^{1-2\unicode[STIX]{x1D702}}\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle & & \displaystyle L^{7/3}N^{9\unicode[STIX]{x1D702}}Z^{1/3}\min \biggl(\frac{q}{Z},\,\frac{N^{1/2+\unicode[STIX]{x1D70C}}}{Z^{1/4}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant L^{7/3}N^{9\unicode[STIX]{x1D702}}q^{1/9}(N^{1/2+\unicode[STIX]{x1D70C}})^{8/9}\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{11/18+\unicode[STIX]{x1D70C}(7/3+1/18+8/9)}\ll N^{1-2\unicode[STIX]{x1D702}},\nonumber\end{eqnarray}$$
completing the discussion for
$k=3$
.
$\mathbf{k\geqslant 4}$
. Using (2.4) and (3.11),
$$\begin{eqnarray}\displaystyle T & \ll & \displaystyle S_{1}^{-1/2+1/k}L^{1-1/k}N^{5\unicode[STIX]{x1D702}}Z^{1/k}\min \biggl(Y,\frac{S_{0}^{1/2}S_{1}q}{Z}\biggr)\nonumber\\ \displaystyle & \ll & \displaystyle L^{1-1/k}N^{5\unicode[STIX]{x1D702}}Y^{1-1/k}S_{0}^{1/2k}S_{1}^{-1/2+2/k}q^{1/k}\nonumber\\ \displaystyle & \ll & \displaystyle N^{\unicode[STIX]{x1D70C}(1-1/k)+(1/2+\unicode[STIX]{x1D70C})(1-1/k)+\unicode[STIX]{x1D70C}/2+1/2+\unicode[STIX]{x1D70C}/2k}\nonumber\\ \displaystyle & \ll & \displaystyle N^{1-2\unicode[STIX]{x1D702}}\nonumber\end{eqnarray}$$
since (as we easily verify)
$$\begin{eqnarray}\unicode[STIX]{x1D70C}\biggl(\frac{5}{2}-\frac{3}{2k}\biggr)<\frac{1}{2k}.\end{eqnarray}$$
This completes the proof of Lemma 8. ◻
4 Type II sums
Our object in the present section is to prove the following.
Lemma 9. For
$k\geqslant 3$
, let
$N^{\unicode[STIX]{x1D70C}}\ll Y\ll N^{1-2J\unicode[STIX]{x1D70C}}$
,
$J=J(f_{k})$
. Let
$|a_{x}|\leqslant 1$
$(x\leqslant N/Y)$
,
$|b_{y}|\leqslant 1$
$(y\sim Y)$
. Then
$$\begin{eqnarray}S:=\mathop{\sum }_{\ell =1}^{L}\Bigg|\sum _{N/2<xy\leqslant N}b_{y}e(\ell g(xy))\Bigg|\ll N^{1-\unicode[STIX]{x1D702}}.\end{eqnarray}$$
We observe that the condition
$N/2<xy\leqslant N$
may be removed at the cost of a log factor [Reference Harman12, §3.2] and we shall show that
$S^{\prime }$
, defined like
$S$
without this condition, is
$\ll N^{1-2\unicode[STIX]{x1D702}}$
.
Proof of Lemma 9.
We write, throughout this section,
$$\begin{eqnarray}S^{\prime }=\mathop{\sum }_{\ell =1}^{L}c_{\ell }\mathop{\sum }_{y\sim Y}b_{y}\mathop{\sum }_{x\leqslant N/Y}a_{x}e(\ell g(xy)),\end{eqnarray}$$
where
$|c_{\ell }|=1$
, so that
$$\begin{eqnarray}|S^{\prime }|\leqslant \mathop{\sum }_{x\leqslant N/Y}\Bigg|\mathop{\sum }_{\ell =1}^{L}c_{\ell }\mathop{\sum }_{y\sim Y}b_{y}e(\ell g(xy))\Bigg|.\end{eqnarray}$$
By the Cauchy–Schwarz inequality,
$$\begin{eqnarray}\displaystyle |S^{\prime }|^{2} & {\leqslant} & \displaystyle \frac{N}{Y}~\mathop{\sum }_{x\leqslant N/Y}\Bigg|\mathop{\sum }_{\ell =1}^{L}~\mathop{\sum }_{y\sim Y}~c_{\ell }b_{y}e(\ell g(xy))\Bigg|^{2}\nonumber\\ \displaystyle & = & \displaystyle \frac{N}{Y}~\mathop{\sum }_{\ell _{1},\ell _{2}=1}^{L}~\mathop{\sum }_{y_{1},y_{2}\sim Y}c_{\ell _{1}}\bar{c}_{\ell _{2}}b_{y_{1}}\bar{b}_{y_{2}}\mathop{\sum }_{x\leqslant N/Y}e(\ell _{1}g(xy_{1})-\ell _{2}g(xy_{2})).\qquad\end{eqnarray}$$
The contribution from quadruples
$(\ell _{1},\ell _{2},y_{1},y_{2})$
with
$\ell _{1}y_{1}^{k}=\ell _{2}y_{2}^{k}$
is
$$\begin{eqnarray}\ll \biggl(\frac{N}{Y}\biggr)^{2}N^{\unicode[STIX]{x1D702}}LY\ll N^{2-4\unicode[STIX]{x1D702}}\end{eqnarray}$$
by a divisor argument since
$Y\geqslant N^{\unicode[STIX]{x1D70C}}$
. Hence, it suffices to show that
$$\begin{eqnarray}\Bigg|\mathop{\sum }_{x\leqslant N/Y}e(\ell _{1}g(xy_{1})-\ell _{2}g(xy_{2}))\Bigg|<\frac{N^{1-4\unicode[STIX]{x1D702}}}{Y}\,L^{-2}\end{eqnarray}$$
for a given quadruple with
$\ell _{1}y_{1}^{k}\neq \ell _{2}y_{2}^{k}$
,
$\ell _{j}\leqslant L$
,
$y_{j}\sim Y$
.
Suppose the contrary. We may apply Lemma 5 with
$X=N/Y$
,
$M=1$
, and
$P=(N^{1-4\unicode[STIX]{x1D702}}/Y)L^{-2}$
. We have
$$\begin{eqnarray}P\geqslant X^{1-1/J+\unicode[STIX]{x1D702}}\end{eqnarray}$$
since
$$\begin{eqnarray}X^{1-1/J+\unicode[STIX]{x1D702}}P^{-1}\leqslant N^{5\unicode[STIX]{x1D702}}\biggl(\frac{N}{Y}\biggr)^{-1/J}L^{2}\leqslant N^{5\unicode[STIX]{x1D702}}(N^{2J\unicode[STIX]{x1D70C}})^{-1/J}L^{2}\leqslant 1.\end{eqnarray}$$
Hence, there exist a natural number
$s$
and an integer
$u$
,
$$\begin{eqnarray}\displaystyle \displaystyle & s\ll (N^{2\unicode[STIX]{x1D702}}L^{2})^{k}\ll N^{2k\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D702}}, & \displaystyle \nonumber\\ \displaystyle \displaystyle & \bigg|s\biggl(\frac{\ell _{1}ay_{1}^{k}-\ell _{2}ay_{2}^{k}}{q}\biggr)-u\bigg|<N^{2k\unicode[STIX]{x1D70C}}\biggl(\frac{N}{Y}\biggr)^{-k}, & \displaystyle \nonumber\end{eqnarray}$$
that is,
$$\begin{eqnarray}\biggl\|\frac{sa}{q}\,(\ell _{1}y_{1}^{k}-\ell _{2}y_{2}^{k})\biggr\|<N^{2k\unicode[STIX]{x1D70C}}\biggl(\frac{N}{Y}\biggr)^{-k}.\end{eqnarray}$$
The right-hand side of (4.2) is less than
$1/q$
since
$Y\ll N^{1/5}$
and
$$\begin{eqnarray}N^{2k\unicode[STIX]{x1D70C}}\biggl(\frac{N}{Y}\biggr)^{-k}q\ll N^{(2k+1/2)\unicode[STIX]{x1D70C}-3k/10}\ll N^{-\unicode[STIX]{x1D702}}\end{eqnarray}$$
(it is easy to verify that
$\unicode[STIX]{x1D70C}<3k/(20k+5)$
). However, the integer
$sa(\ell _{1}y_{1}^{k}-\ell _{2}y_{2}^{k})$
is not divisible by
$q$
since
$$\begin{eqnarray}1\leqslant |s(\ell _{1}y_{1}^{k}-\ell _{2}y_{2}^{k})|\ll N^{(2k+1)\unicode[STIX]{x1D70C}}Y^{k}\ll qN^{-\unicode[STIX]{x1D702}}\end{eqnarray}$$
by the same inequality
$$\begin{eqnarray}\biggl(2k+\frac{1}{2}\biggr)\unicode[STIX]{x1D70C}\leqslant \frac{3k}{10}-\unicode[STIX]{x1D702}\end{eqnarray}$$
used in (4.3). Thus, (4.2) cannot hold. This completes the proof of Lemma 9. ◻
Before we consider a variant of Lemma 9 for
$k=2$
, we note the following way of using the Cauchy–Schwarz inequality:
$$\begin{eqnarray}\displaystyle |S^{\prime }|^{2} & {\leqslant} & \displaystyle \Bigg\{\mathop{\sum }_{\ell =1}^{L}\mathop{\sum }_{x\leqslant N/Y}\Bigg|\mathop{\sum }_{y\sim Y}b_{y}e(\ell g(xy))\Bigg|\Bigg\}^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{LN}{Y}~\mathop{\sum }_{y_{1},y_{2}\sim Y}\Bigg|\mathop{\sum }_{\ell =1}^{L}~\mathop{\sum }_{x\leqslant N/Y}e(\ell (g(xy_{1})-g(xy_{2})))\Bigg|.\nonumber\end{eqnarray}$$
To prove that
$|S^{\prime }|^{2}\ll N^{2-4\unicode[STIX]{x1D702}}$
, it suffices to show that
$$\begin{eqnarray}R:=\mathop{\sum }_{y_{1},y_{2}\sim Y}~\mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{x\leqslant N/Y}e(\ell (g(xy_{1})-g(xy_{2})))\Bigg|\ll N^{1-4\unicode[STIX]{x1D702}}YL^{-1}.\end{eqnarray}$$
We need one more lemma.
Lemma 10. Let
$W$
,
$X$
,
$Y$
be positive integers greater than 1. Then the inequality
$$\begin{eqnarray}\min _{s\leqslant W}\biggl\|\frac{a(y_{1}^{2}-y_{2}^{2})s}{q}\biggr\|<\frac{1}{X}\end{eqnarray}$$
is satisfied for
$$\begin{eqnarray}\ll \biggl(\frac{WY^{2}}{q}+1\biggr)\biggl(1+\frac{q}{X}\biggr)(WY)^{\unicode[STIX]{x1D702}}\end{eqnarray}$$
pairs
$y_{1}$
,
$y_{2}$
with
$y_{1}\neq y_{2}$
,
$y_{1}$
,
$y_{2}\sim Y$
.
Proof. See [Reference Baker and Harman4, Lemma 7]. ◻
Lemma 11. Let
$k=2$
. Let
$N^{2\unicode[STIX]{x1D70C}}\ll Y\ll N^{1-4\unicode[STIX]{x1D70C}}$
. Then, for
$|a_{x}|\leqslant 1$
$(x\leqslant N/Y)$
,
$|b_{y}|\leqslant 1$
$(y\sim Y)$
, we have
$$\begin{eqnarray}\mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{x\leqslant N/Y}\,a_{x}\mathop{\sum }_{\substack{ y\sim Y \\ N/2<xy\leqslant N}}b_{y}e(\ell g(xy))\Bigg|\ll N^{1-\unicode[STIX]{x1D702}}.\end{eqnarray}$$
Proof. As already noted, it suffices to prove (4.4). Since
$Y\gg L^{2}N^{4\unicode[STIX]{x1D702}}$
, we need only consider the contribution to
$R$
from pairs
$(y_{1},y_{2})$
with
$y_{1}\neq y_{2}$
,
$$\begin{eqnarray}\mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{x\leqslant N/Y}e(\ell (g(xy_{1})-g(xy_{2})))\Bigg|>N^{1-4\unicode[STIX]{x1D702}}Y^{-1}L^{-1}.\end{eqnarray}$$
For such a pair
$(y_{1},y_{2})$
, we apply Lemma 5, with
$k=2$
;
$L$
,
$N/Y$
in place of
$M$
,
$X$
; and
$P=N^{1-4\unicode[STIX]{x1D702}}Y^{-1}L^{-1}$
. We have
$$\begin{eqnarray}P\geqslant L\biggl(\frac{N}{Y}\biggr)^{1/2+\unicode[STIX]{x1D702}}\end{eqnarray}$$
since
$Y\ll N^{1-4\unicode[STIX]{x1D70C}}$
. Thus (suppressing dependence on
$y_{1}$
,
$y_{2}$
), there are natural numbers
$s$
and integers
$u_{1}$
,
$u_{2}$
with
$s\ll N^{4\unicode[STIX]{x1D70C}}$
,
$\langle s,u_{2}\rangle \leqslant LN^{\unicode[STIX]{x1D702}}$
and, for
$\unicode[STIX]{x1D6FE}_{2}=(y_{1}^{2}-y_{2}^{2})\,a/q$
,
$\unicode[STIX]{x1D6FE}_{1}=(y_{1}-y_{2})\unicode[STIX]{x1D6FC}_{1}$
, satisfying
$$\begin{eqnarray}|s\unicode[STIX]{x1D6FE}_{j}-u_{j}|\ll L^{-1}\biggl(\frac{N}{Y}\biggr)^{-j}N^{4\unicode[STIX]{x1D70C}-2\unicode[STIX]{x1D702}}\quad (j=1,2).\end{eqnarray}$$
It is clear that Lemma 7 is applicable. Thus,
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{x\leqslant N/Y}e(\ell (\unicode[STIX]{x1D6FE}_{2}x^{2}+\unicode[STIX]{x1D6FE}_{1}x))-s^{-1}S(s,\ell G)\int _{0}^{N/Y}e(\ell F(z))\,dz\Bigg|\nonumber\\ \displaystyle & & \displaystyle \qquad \ll N^{2\unicode[STIX]{x1D702}}Ls^{1/2}\ll N^{3\unicode[STIX]{x1D70C}-5\unicode[STIX]{x1D702}}\ll \frac{N^{1-5\unicode[STIX]{x1D702}}}{YL}.\nonumber\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\ell =1}^{L}\Bigg|\mathop{\sum }_{x\leqslant N/Y}e(\ell (\unicode[STIX]{x1D6FE}_{2}x^{2}+\unicode[STIX]{x1D6FE}_{1}x))\Bigg| & \ll & \displaystyle \mathop{\sum }_{\ell =1}^{L}s^{-1/2}\langle s,\ell \rangle ^{1/2}\frac{N}{Y}\nonumber\\ \displaystyle & \ll & \displaystyle Ls^{-1/2}\,\frac{N^{1+\unicode[STIX]{x1D702}}}{Y}.\end{eqnarray}$$
In particular, for these pairs
$(y_{1},y_{2})$
, we have
$$\begin{eqnarray}s^{1/2}\ll L^{2}N^{5\unicode[STIX]{x1D702}};\qquad 2s\leqslant N^{4\unicode[STIX]{x1D70C}}.\end{eqnarray}$$
Let
$X=Y^{-2}LN^{2-4\unicode[STIX]{x1D70C}}$
. Then (4.5) implies
$$\begin{eqnarray}\biggl\|\frac{sa(y_{1}^{2}-y_{2}^{2})}{q}\biggr\|<\frac{1}{X}.\end{eqnarray}$$
The number of pairs
$(y_{1},y_{2})$
with
$s\sim W$
(for
$W\leqslant N^{4\unicode[STIX]{x1D70C}}$
) is
$$\begin{eqnarray}\ll \biggl(\frac{WY^{2}}{q}+1\biggr)\biggl(1+\frac{q}{X}\biggr)N^{\unicode[STIX]{x1D702}}\end{eqnarray}$$
by Lemma 10 and these pairs contribute to
$R$
an amount
$$\begin{eqnarray}\displaystyle \ll \frac{LN^{1+2\unicode[STIX]{x1D702}}}{YW^{1/2}}\biggl(\frac{WY^{2}}{q}+1\biggr)\biggl(1+\frac{q}{X}\biggr) & & \displaystyle \nonumber\end{eqnarray}$$
(by (4.6))
$$\begin{eqnarray}\displaystyle \ll \frac{W^{1/2}LN^{1+2\unicode[STIX]{x1D702}}Y}{q}+\frac{W^{1/2}LN^{1+2\unicode[STIX]{x1D702}}Y}{X}+\frac{LN^{1+2\unicode[STIX]{x1D702}}}{Y}+\frac{LN^{1+2\unicode[STIX]{x1D702}}q}{XY}. & & \displaystyle \nonumber\end{eqnarray}$$
Each of these four terms is
$\ll N^{1-5\unicode[STIX]{x1D702}}YL^{-1}$
:
$$\begin{eqnarray}\displaystyle \frac{W^{1/2}LN^{1+2\unicode[STIX]{x1D702}}Y}{q} & \ll & \displaystyle N^{5\unicode[STIX]{x1D70C}/2}Y\ll N^{1-5\unicode[STIX]{x1D702}}YL^{-1}\quad (\text{since }\unicode[STIX]{x1D70C}<2/7);\nonumber\\ \displaystyle \frac{W^{1/2}LN^{1+2\unicode[STIX]{x1D702}}Y}{X} & = & \displaystyle W^{1/2}N^{-1+4\unicode[STIX]{x1D70C}+2\unicode[STIX]{x1D702}}Y^{3}\ll N^{-1+6\unicode[STIX]{x1D70C}+2\unicode[STIX]{x1D702}}Y^{3}\ll \frac{N^{1-5\unicode[STIX]{x1D702}}Y}{L}\nonumber\\ \displaystyle & & \displaystyle \nonumber\end{eqnarray}$$
(since
$Y\ll N^{1-4\unicode[STIX]{x1D70C}}$
);
$$\begin{eqnarray}\displaystyle \frac{LN^{1+2\unicode[STIX]{x1D702}}}{Y}\ll \frac{N^{1-5\unicode[STIX]{x1D702}}Y}{L}\quad (\text{since }Y\gg N^{2\unicode[STIX]{x1D70C}}); & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \frac{LN^{1+2\unicode[STIX]{x1D702}}q}{XY}=qN^{-1+4\unicode[STIX]{x1D70C}+2\unicode[STIX]{x1D702}}Y\ll N^{9\unicode[STIX]{x1D70C}/2+2\unicode[STIX]{x1D702}}Y\ll \frac{N^{1-5\unicode[STIX]{x1D702}}Y}{L} & & \displaystyle \nonumber\end{eqnarray}$$
(since
$\unicode[STIX]{x1D70C}<2/11$
).
We now sum over
$O(\log N)$
values of
$W=N^{4\unicode[STIX]{x1D70C}}2^{-j}$
and obtain the desired bound (4.4). This completes the proof of Lemma 11.◻
5 Application of the Harman sieve
We use the standard notation
$$\begin{eqnarray}\displaystyle P(z) & = & \displaystyle \mathop{\prod }_{p<z}p,\nonumber\\ \displaystyle E_{d} & = & \displaystyle \{n:dn\in E\}\quad \text{for a finite subset }E\text{ of }\mathbb{N},\text{ while}\nonumber\\ \displaystyle S(E,z) & = & \displaystyle \mathop{\sum }_{\substack{ n\in E \\ \langle n,P(z)\rangle =1}}1,\qquad \unicode[STIX]{x1D712}_{E}=\text{indicator function of }E.\nonumber\end{eqnarray}$$
We “compare” the set
$A$
introduced in §1,
$(A=A(f_{k}),J=J(f_{k}))$
with the set
$$\begin{eqnarray}B=\biggl\{n\in \mathbb{N}:\frac{N}{2}<n\leqslant N\biggr\}.\end{eqnarray}$$
We write
$[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}]=[2\unicode[STIX]{x1D70C},1-5\unicode[STIX]{x1D70C}/2]$
$(k=2)$
;
$[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}]=[\unicode[STIX]{x1D70C},1-2J\unicode[STIX]{x1D70C}]$
$(k>2)$
.
Lemma 12. Let
$u_{h}(h\leqslant H)$
be real numbers with
$|u_{h}|\leqslant 1$
,
$u_{h}=0$
for
$(h,P(N^{\unicode[STIX]{x1D702}}))>1$
. Suppose that
$$\begin{eqnarray}H<MN^{-\unicode[STIX]{x1D6FC}},\end{eqnarray}$$
where
$M\ll N^{1-5\unicode[STIX]{x1D70C}/2}$
for
$k=2$
and
$M\ll N^{1/2+\unicode[STIX]{x1D70C}}$
for
$k\geqslant 3$
. Then, writing
$\unicode[STIX]{x1D6FF}$
for
$L_{1}^{-1}$
,
$$\begin{eqnarray}\mathop{\sum }_{h\leqslant H}u_{h}S(A_{h},N^{\unicode[STIX]{x1D6FD}})-2\unicode[STIX]{x1D6FF}\mathop{\sum }_{h\leqslant H}u_{h}S(B_{h},N^{\unicode[STIX]{x1D6FD}})\ll \unicode[STIX]{x1D6FF}N^{1-\unicode[STIX]{x1D702}/2}.\end{eqnarray}$$
Proof. We first apply [Reference Baker and Weingartner5, Lemma 14], which is a variant of [Reference Harman12, Theorem 3.1] convenient for our purposes. By choosing the weight function of the lemma to be
$$\begin{eqnarray}w(n)=\unicode[STIX]{x1D712}_{A}(n)-2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}_{B}(n),\end{eqnarray}$$
we see that we need only show that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ N/2<mn\leqslant N \\ m\leqslant M}}a_{m}(\unicode[STIX]{x1D712}_{A}(mn)-2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}_{B}(mn)) & \ll & \displaystyle \unicode[STIX]{x1D6FF}N^{1-2\unicode[STIX]{x1D702}/3},\nonumber\\ \displaystyle \mathop{\sum }_{N^{\unicode[STIX]{x1D6FC}}\leqslant m\leqslant N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}}a_{m}\mathop{\sum }_{n}b_{n}(\unicode[STIX]{x1D712}_{A}(mn)-2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}_{B}(mn)) & \ll & \displaystyle \unicode[STIX]{x1D6FF}N^{1-2\unicode[STIX]{x1D702}/3},\nonumber\end{eqnarray}$$
where the
$a_{m}$
,
$b_{n}$
are arbitrary with
$|a_{m}|\leqslant 1$
,
$|b_{n}|\leqslant 1$
. Now a standard use of upper and lower bounds for the indicator function of
$\unicode[STIX]{x1D712}_{A}$
reduces our task to proving bounds for exponential sums that have already been given in Lemmas 8 and 9; compare, for example, the arguments in [Reference Harman12, §3.4].◻
Another “comparison” that will be needed below is a bound
$O(\unicode[STIX]{x1D6FF}N^{1-2\unicode[STIX]{x1D702}/3})$
for a sum
$$\begin{eqnarray}\mathop{\sum }_{N^{\unicode[STIX]{x1D6FC}}\leqslant p\leqslant N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}}(S(A_{p},p)-2\unicode[STIX]{x1D6FF}S(B_{p},p)).\end{eqnarray}$$
This reduces to a sum of
$O(1)$
expressions, in which
$r=O(1)$
, of the form
$$\begin{eqnarray}\mathop{\sum }_{\substack{ N^{\unicode[STIX]{x1D6FC}}\leqslant p\leqslant N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}},\,N/2<pp_{1}\ldots p_{r}\leqslant N \\ p\leqslant p_{1}\leqslant p_{2}\leqslant \cdots \leqslant p_{r}}}\{\unicode[STIX]{x1D712}_{A}(pp_{1}\ldots p_{r})-2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D712}_{B}(pp_{1}\ldots p_{r})\}.\end{eqnarray}$$
Lemma 9 provides a satisfactory estimate, using [Reference Harman12, §3.2] to remove the condition
$p\leqslant p_{1}$
that will occur in the Type II exponential sums that arise.
Proof of Theorem 1 for
$k\geqslant 3$
.
We observe that
$$\begin{eqnarray}\displaystyle \#\{p:p\in A\} & = & \displaystyle S(A,(2N)^{1/2})\nonumber\\ \displaystyle & = & \displaystyle S(A,N^{\unicode[STIX]{x1D6FC}})-\mathop{\sum }_{N^{\unicode[STIX]{x1D6FC}}\leqslant p\leqslant (2N)^{1/2}}S(A_{p},p)\end{eqnarray}$$
by an application of Buchstab’s identity. Iterating the procedure,
$$\begin{eqnarray}\#\{p:p\in A\}=S_{1}-S_{2}-S_{3}+S_{4},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \displaystyle S_{1}=S(A,N^{\unicode[STIX]{x1D6FC}}),\qquad S_{2}=\mathop{\sum }_{N^{\unicode[STIX]{x1D6FC}}\leqslant p\leqslant N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}}S(A_{p},p), & \displaystyle \nonumber\\ \displaystyle & \displaystyle S_{3}=\mathop{\sum }_{N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}<p\leqslant (2N)^{1/2}}\biggl(S(A_{p},N^{\unicode[STIX]{x1D6FD}})-\mathop{\sum }_{N^{\unicode[STIX]{x1D6FD}}\leqslant q<N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}}S(A_{pq},q)\biggr), & \displaystyle \nonumber\\ \displaystyle & \displaystyle S_{4}=\mathop{\sum }_{\substack{ N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}<p\leqslant (2N)^{1/2} \\ N^{\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}}\leqslant q<p}}S(A_{pq},q). & \displaystyle \nonumber\end{eqnarray}$$
Define
$S_{1}^{\prime }$
,
$S_{2}^{\prime }$
,
$S_{3}^{\prime }$
, and
$S_{4}^{\prime }$
in the same way as
$S_{1},S_{2},S_{3}$
, and
$S_{4}$
, with
$A$
replaced by
$B$
. We observe that
$$\begin{eqnarray}\displaystyle \#\{p:p\in A\} & {\geqslant} & \displaystyle S_{1}-S_{2}-S_{3}\nonumber\\ \displaystyle & = & \displaystyle 2\unicode[STIX]{x1D6FF}(S_{1}^{\prime }-S_{2}^{\prime }-S_{3}^{\prime })+O(\unicode[STIX]{x1D6FF}N^{1-\unicode[STIX]{x1D702}/2})\nonumber\end{eqnarray}$$
by applying Lemma 12, and the remarks following the proof of Lemma 12, to
$S_{1}$
,
$S_{2}$
and
$S_{3}$
. (The condition (5.1) will be trivial for
$S_{1}$
and will amount to
$$\begin{eqnarray}(2N)^{1/2}\leqslant MN^{-\unicode[STIX]{x1D70C}}\end{eqnarray}$$
for
$S_{3}$
, where
$M=2^{1/2}N^{1/2+\unicode[STIX]{x1D70C}}$
.) We now follow arguments familiar from [Reference Harman12]. To show that
$S_{1}^{\prime }-S_{2}^{\prime }-S_{3}^{\prime }>b(N\unicode[STIX]{x1D6FF}/\text{log}\,N)$
for a small positive
$b$
, it suffices to show that,
$\unicode[STIX]{x1D714}$
denoting the Buchstab function,
$$\begin{eqnarray}\iint _{0.1842<y<x<1/2,\,x+2y<1}\unicode[STIX]{x1D714}\biggl(\frac{1-x-y}{y}\biggr)\frac{dx}{x}~\frac{dy}{y^{2}}<1.\end{eqnarray}$$
(Here we use
$\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=2/10$
for
$k=3$
and
$\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}=0.1842$
for
$k\geqslant 4$
.) The constant 0.1842 is close to sharp (within
$10^{-4}$
). The validity of (5.3) was kindly checked by Andreas Weingartner.◻
Proof of Theorem 1 for
$k=2$
.
We now have
$$\begin{eqnarray}[\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}]=\big[{\textstyle \frac{4}{13}},{\textstyle \frac{5}{13}}\big]\end{eqnarray}$$
and the condition (5.1) reduces to
$$\begin{eqnarray}H\ll N^{8/13}.\end{eqnarray}$$
We can weaken this restriction on
$H$
to
$$\begin{eqnarray}H\ll N^{9/13}\end{eqnarray}$$
by treating the Type I exponential sums in question as Type II exponential sums with one variable between
$N^{1-5/13}$
and
$O(N^{1-4/13})$
. Now we have stronger “arithmetic information” than is used by Harman [Reference Harman12, §5.3] in the discussion of the Diophantine inequality.
$$\begin{eqnarray}\Vert p\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\Vert <p^{-0.3182}\quad (\text{since }{\textstyle \frac{2}{13}}<{\textstyle \frac{0.3182}{2}}).\end{eqnarray}$$
We can follow the proof there verbatim to obtain Theorem 1.◻
