1 Introduction
The subject of metric Diophantine approximation of manifolds studies Diophantine approximation of typical points on submanifolds in
$\mathbb{R}^{d}$
by rational points in
$\mathbb{R}^{d}$
. This subject has received considerable attention in the last two decades, leading to dramatic progress using methods arising from the ergodic theory of flows on homogeneous spaces as well as analytic methods. If one puts further restrictions on the approximating rationals, then the situation is much less well understood. A very natural class of problems arises by imposing primality restrictions on the approximating rationals. In this paper, we study the problem of Diophantine approximation for vectors on lines in
$\mathbb{R}^{d}$
with additional primality restrictions. Thus we combine the themes of simultaneous metric Diophantine approximation on affine subspaces and Diophantine approximation with restrictions. Both problems have their own substantial complications and have separately received considerable attention (cf. [Reference Beresnevich, Bernik, Dickinson and Dodson1, Reference Ghosh5, Reference Ghosh6, Reference Kleinbock15] for Diophantine approximation on affine subspaces and [Reference Harman9–Reference Harman11, Reference Heath-Brown and Jia13, Reference Jones14, Reference Matomäki16–Reference Srinivasan18] for Diophantine approximation with primality constraints). The only previous works on the combined theme that we are aware of are the work of Harman and Jones [Reference Harman and Jones12], regarding Diophantine approximation on curves in
$\mathbb{R}^{2}$
with constraints, and our previous work [Reference Baier and Ghosh2], where we addressed the problem of Diophantine approximation with constraints on lines in
$\mathbb{R}^{2}$
. The main result of the present paper generalises [Reference Baier and Ghosh2] to arbitrary dimensions and is the first such result in this generality. While the broad strategy in the present paper is similar to that of [Reference Baier and Ghosh2], the greater generality makes the problem significantly more complicated. In studying Diophantine approximation on lines and, more generally, affine subspaces, it is natural and indeed imperative to impose some Diophantine condition on the line or subspace itself. In [Reference Baier and Ghosh2], we had assumed that the slope of the line in
$\mathbb{R}^{2}$
is irrational. In higher dimensions, the lack of a suitable continued fraction algorithm makes it necessary to replace irrationality with a suitable Diophantine condition, which we now introduce. Let
$\Vert ~\Vert$
denote the distance to the nearest integer of a real number, let
$\Vert ~\Vert _{\infty }$
denote the supremum norm of a vector and, for vectors
$\mathbf{v},\mathbf{c}\in \mathbb{R}^{d}$
, denote by
$\mathbf{v}\cdot \mathbf{c}=v_{1}c_{1}+\cdots +v_{d}c_{d}$
the inner product of
$\mathbf{v}$
and
$\mathbf{c}$
. Recall that
$\mathbf{c}\in \mathbb{R}^{d}$
is called
$k$
-Diophantine (
$\mathbf{c}\in D_{k}(\mathbb{R}^{d})$
) if there exists a constant
$C>0$
such that
$$\begin{eqnarray}\Vert \mathbf{v}\cdot \mathbf{c}\Vert >\frac{C}{\Vert \mathbf{v}\Vert _{\infty }^{k}}\quad \text{for every }\mathbf{v}\in \mathbb{Z}^{d}\setminus \{\mathbf{0}\}.\end{eqnarray}$$
Our main result is the following theorem.
Theorem 1.1. Let
$d$
be a positive integer and let
$k\geqslant d$
be a positive real number. Define
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{d,k}:=\frac{1}{d(3k+2)}\end{eqnarray}$$
and suppose that
$0<\unicode[STIX]{x1D700}<\unicode[STIX]{x1D6FE}_{d,k}$
. Let
$c_{1},\ldots ,c_{d}$
be positive irrational numbers such that the vector
$\mathbf{c}=(c_{1},\ldots ,c_{d})$
is
$k$
-Diophantine. Then, for almost all positive real
$\unicode[STIX]{x1D6FC}$
, with respect to the Lebesgue measure, there are an infinite number of
$(d+2)$
-tuples
$(p,q_{1},\ldots ,q_{d},r)$
with
$p$
and
$r$
prime and
$q_{1},\ldots ,q_{d}$
positive integers such that, simultaneously,
$$\begin{eqnarray}\begin{array}{@{}l@{}}0<p\unicode[STIX]{x1D6FC}-r<p^{-\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}},\\ 0<pc_{i}\unicode[STIX]{x1D6FC}-q_{i}<p^{-\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}}\quad \text{for all }i\in \{1,\ldots ,d\}.\end{array}\end{eqnarray}$$
Remarks.
(1) It is well known that
$D_{d}(\mathbb{R}^{d})$
is a non-empty set of zero Lebesgue measure and full Hausdorff dimension. These comprise the set of badly approximable vectors. Moreover,
$D_{k}(\mathbb{R}^{d})$
has full measure whenever
$k>d$
(see [Reference Cassels4], for example).
(2) In [Reference Baier and Ghosh2], we proved the analogue of Theorem 1.1 for lines in
$\mathbb{R}^{2}$
under the assumption that the slope
$c$
of the line is irrational. In fact, what was used was the following Diophantine property of irrational numbers: there exists an infinite set
${\mathcal{S}}$
of integers and a positive constant
$D>0$
such that
$$\begin{eqnarray}\min _{\substack{ \mathbf{v}\in \mathbb{Z}^{d}\setminus \{\mathbf{0}\} \\ \Vert \mathbf{v}\Vert _{\infty }\leqslant N}}\Vert \mathbf{v}\cdot \mathbf{c}\Vert \geqslant N^{-D}\quad \text{if }N\in {\mathcal{S}}.\end{eqnarray}$$
For
$d=1$
, this holds for
$D=1$
and for
${\mathcal{S}}$
being the set of numbers
$[N/2]$
, where the
$N$
are the denominators in the continued fraction approximants of
$c=c_{1}$
. This can be seen by approximating
$c$
by its continued fraction approximant
$a/N$
and using the fact that
$|c-a/N|\leqslant 1/N^{2}$
. In the case
$d=1$
, a sequence
${\mathcal{S}}$
with the above property can also be constructed using the Dirichlet approximation theorem and the condition that
$c$
is irrational. There is a
$d$
-dimensional version of Dirichlet’s approximation theorem, but for
$d\geqslant 2$
, a statement such as (4) does not follow from it. It is likely that the conclusion of Theorem 1.1 would also hold in arbitrary dimension by assuming (4) rather than the Diophantine condition that we have assumed, with perhaps a different exponent. We note, however, that, for
$d=1$
, by choosing
$k=1$
we recover the exponent in [Reference Baier and Ghosh2]. The condition (4) is also an interesting Diophantine property, and it can be shown to include the class of non-singular vectors.
(3) It is an interesting problem to consider analogues of Theorem 1.1 for affine subspaces of lower codimension and, indeed, for manifolds not contained in affine subspaces, that is, the non-degenerate manifolds. We will consider these in a forthcoming work.
2 A metrical approach
Our method is based on the following lemma in [Reference Harman and Jones12].
Lemma 2.1 [Reference Harman and Jones12, Lemma 1].
Let
$A$
and
$B$
be reals with
$B>A>0$
. Let
$F_{N}(\unicode[STIX]{x1D6FC})$
be a non-negative valued function of N (an integer) and
$\unicode[STIX]{x1D6FC}$
(a real variable) and let
$G_{N}$
,
$V_{N}$
be functions of N such that:
-
(i)
$G_{N}\rightarrow \infty$
as
$N\rightarrow \infty$
; -
(ii)
$V_{N}=o(G_{N})$
as
$N\rightarrow \infty$
; -
(iii) for all
$a$
,
$b$
with
$A\leqslant a<b\leqslant B$
,
$$\begin{eqnarray}\limsup _{N\rightarrow \infty }\int _{a}^{b}\frac{F_{N}(\unicode[STIX]{x1D6FC})}{G_{N}}\,d\unicode[STIX]{x1D6FC}\geqslant b-a;\end{eqnarray}$$
-
(iv) there is a positive constant K such that, for any measurable set
${\mathcal{C}}\subseteq$
$[A,B]$
,
$$\begin{eqnarray}\int _{{\mathcal{C}}}F_{N}(\unicode[STIX]{x1D6FC})\,d\unicode[STIX]{x1D6FC}\leqslant KG_{N}\unicode[STIX]{x1D706}({\mathcal{C}})+V_{N}.\end{eqnarray}$$
Then, for almost all
$\unicode[STIX]{x1D6FC}\in [A,B]$
, (5)
$$\begin{eqnarray}\limsup _{N\rightarrow \infty }\frac{F_{N}(\unicode[STIX]{x1D6FC})}{G_{N}}\geqslant 1.\end{eqnarray}$$
In our application,
$F_{N}(\unicode[STIX]{x1D6FC})$
will be the number of solutions to (3) with
$p<N$
. Further, for given
$0<A<B$
we set
$$\begin{eqnarray}G_{N}=G_{N}(A,B)=\frac{A^{2}}{B^{2}}\cdot \frac{\min (c_{1},\ldots ,c_{d},d)^{d-1}}{2^{d+1}}N^{1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}(\log N)^{-2}.\end{eqnarray}$$
We will prove the following theorem.
Theorem 2.2. The following holds for every natural number
$N$
.
-
(i) Let
$0<A<B$
. Then, for all
$a,b$
with
$A\leqslant a<b\leqslant B$
, (6)if
$$\begin{eqnarray}\int _{a}^{b}F_{N}(\unicode[STIX]{x1D6FC})\,d\unicode[STIX]{x1D6FC}\geqslant (b-a)G_{N}(A,B)(1+o(1))\end{eqnarray}$$
$N\in {\mathcal{S}}$
and
$N\rightarrow \infty$
.
-
(ii) Let
$0<A<B$
and
$\unicode[STIX]{x1D700}>0$
. Then there exists a constant
$K=K(A,B,\unicode[STIX]{x1D700})$
such that, for
$\unicode[STIX]{x1D6FC}\in [A,B]$
, with
$$\begin{eqnarray}F_{N}(\unicode[STIX]{x1D6FC})\leqslant KG_{N}(A,B)+J_{N}(\unicode[STIX]{x1D6FC})\end{eqnarray}$$
if
$$\begin{eqnarray}\int _{A}^{B}|J_{N}(\unicode[STIX]{x1D6FC})|\,d\unicode[STIX]{x1D6FC}=o(G_{N}(A,B))\quad \text{as }N\rightarrow \infty\end{eqnarray}$$
$N\in {\mathcal{S}}$
and
$N\rightarrow \infty$
.
Theorem 2.2(i) corresponds to Lemma 2 and Theorem 2.2(ii) corresponds to Lemma 3 in [Reference Harman and Jones12]. Now it follows that conditions (i) to (iv) in Lemma 2.1 are satisfied for
$$\begin{eqnarray}V_{N}=\int _{A}^{B}|J_{N}(\unicode[STIX]{x1D6FC})|\,d\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
3 Proof of Theorem 2.2(i)
3.1 Reduction to a counting problem
To prove Theorem 2.2(i), we broadly follow the approach in [Reference Harman and Jones12, §3]. However, we use exponential sum estimates instead of zero density estimates for the Riemann zeta function since they turn out to be more suitable for our purposes. This is the content of the next subsection.
Throughout the following, we denote the set of primes by
$\mathbb{P}$
. Let
$$\begin{eqnarray}\displaystyle {\mathcal{B}}_{p} & = & \displaystyle \mathop{\bigcup }_{\substack{ r\in \mathbb{P} \\ q_{1},\ldots ,q_{d}\in \mathbb{N}}}\biggl[\frac{r}{p},\frac{r+\tilde{\unicode[STIX]{x1D702}}}{p}\biggr)\cap \biggl[\frac{1}{c_{1}}\cdot \frac{q_{1}}{p},\frac{1}{c_{1}}\cdot \frac{q_{1}+\tilde{\unicode[STIX]{x1D702}}}{p}\biggr)\nonumber\\ \displaystyle & & \displaystyle \cap \cdots \cap \biggl[\frac{1}{c_{d}}\cdot \frac{q_{d}}{p},\frac{1}{c_{d}}\cdot \frac{q_{d}+\tilde{\unicode[STIX]{x1D702}}}{p}\biggr)\cap [a,b],\nonumber\end{eqnarray}$$
where
$\tilde{\unicode[STIX]{x1D702}}=p^{\unicode[STIX]{x1D700}-\unicode[STIX]{x1D6FE}_{d,k}}$
. Then
$$\begin{eqnarray}\int _{a}^{b}F_{N}(\unicode[STIX]{x1D6FC})\,d\unicode[STIX]{x1D6FC}=\mathop{\sum }_{\substack{ p\in \mathbb{P} \\ p\leqslant N}}\unicode[STIX]{x1D706}({\mathcal{B}}_{p}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D706}$
is Lebesgue measure. Set
$$\begin{eqnarray}\unicode[STIX]{x1D707}:=(a+b)/(2a).\end{eqnarray}$$
Our strategy is to split the interval
$[1,N]$
into subintervals
$[P,P\unicode[STIX]{x1D707}]$
and sum up over the
$P$
. Accordingly, we restrict
$p$
to the interval
$P\leqslant p<P\unicode[STIX]{x1D707}$
with
$P\unicode[STIX]{x1D707}\leqslant N$
. We then obtain a lower bound for (7) by replacing
$\tilde{\unicode[STIX]{x1D702}}$
with
$$\begin{eqnarray}\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D707}P)^{\unicode[STIX]{x1D700}-\unicode[STIX]{x1D6FE}_{d,k}}.\end{eqnarray}$$
Clearly,
$\tilde{\unicode[STIX]{x1D702}}\geqslant \unicode[STIX]{x1D702}$
if
$P\leqslant p<P\unicode[STIX]{x1D707}$
.
We note that if
$$\begin{eqnarray}\frac{r}{p}\leqslant \frac{1}{c_{i}}\cdot \frac{q_{i}}{p}\leqslant \frac{r+\unicode[STIX]{x1D702}/2}{p}\quad \text{for }i=1,\ldots ,d,\end{eqnarray}$$
then
$$\begin{eqnarray}\unicode[STIX]{x1D706}\biggl(\biggl[\frac{r}{p},\frac{r+\unicode[STIX]{x1D702}}{p}\biggr)\cap \biggl[\frac{1}{c_{1}}\cdot \frac{q_{1}}{p},\frac{1}{c_{1}}\cdot \frac{q_{1}+\unicode[STIX]{x1D702}}{p}\biggr)\cap \cdots \cap \biggl[\frac{1}{c_{d}}\cdot \frac{q_{d}}{p},\frac{1}{c_{d}}\cdot \frac{q_{d}+\unicode[STIX]{x1D702}}{p}\biggr)\biggr)\geqslant \unicode[STIX]{x1D708},\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D708}:=\frac{\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D707}P}\min \biggl(\frac{1}{2},\frac{1}{c_{1}},\ldots ,\frac{1}{c_{d}}\biggr)=(\unicode[STIX]{x1D707}P)^{-1-\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}}\min \biggl(\frac{1}{2},\frac{1}{c_{1}},\ldots ,\frac{1}{c_{d}}\biggr).\end{eqnarray}$$
Also, for all
$p\in [P,P\unicode[STIX]{x1D707})$
,
$$\begin{eqnarray}Pa\unicode[STIX]{x1D707}\leqslant r\leqslant bP\Longrightarrow a\leqslant \frac{r}{p}\leqslant b,\end{eqnarray}$$
where
$r$
runs over the primes in an interval of length
$(b-a)/2P$
. Thus
$$\begin{eqnarray}\mathop{\sum }_{P\leqslant p<\unicode[STIX]{x1D707}P}\unicode[STIX]{x1D706}({\mathcal{B}}_{p})\geqslant \unicode[STIX]{x1D708}N(P),\end{eqnarray}$$
where
$N(P)$
counts the number of solutions
$(p,q_{1},\ldots ,q_{d},r)\in \mathbb{P}\times \mathbb{Z}^{d}\times \mathbb{P}$
to
$$\begin{eqnarray}q_{i}\in [c_{i}r,c_{i}r+\unicode[STIX]{x1D6FF})\quad \text{for }i=1,\ldots ,d,\qquad P\leqslant p<P\unicode[STIX]{x1D707},\qquad Pa\unicode[STIX]{x1D707}\leqslant r\leqslant bP,\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}:=\frac{\min \{c_{1},\ldots ,c_{d}\}\unicode[STIX]{x1D702}}{2}=\frac{\min \{c_{1},\ldots ,c_{d}\}}{2(\unicode[STIX]{x1D707}P)^{\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700}}}.\end{eqnarray}$$
Note that, in contrast to the problem considered by Harman and Jones, the conditions on
$p$
and
$q$
are here independent, which simplifies matters to some extent. By the prime number theorem, the number
$R(P)$
of prime solutions to
$$\begin{eqnarray}P\leqslant p<P\unicode[STIX]{x1D707}\end{eqnarray}$$
satisfies
$$\begin{eqnarray}R(P)\sim (\unicode[STIX]{x1D707}-1)P(\log 2P)^{-1}\quad \text{as }P\rightarrow \infty .\end{eqnarray}$$
It remains to count the number of solutions
$(q_{1},\ldots ,q_{d},r)\in \mathbb{N}\times \mathbb{P}$
to
$$\begin{eqnarray}q_{i}\in [c_{i}r,c_{i}r+\unicode[STIX]{x1D6FF})\quad \text{for }i=1,\ldots ,d,\qquad Pa\unicode[STIX]{x1D707}\leqslant r\leqslant bP,\end{eqnarray}$$
which equals
$$\begin{eqnarray}S(P):=\mathop{\sum }_{\substack{ Pa\unicode[STIX]{x1D707}\leqslant r\leqslant bP \\ r\,\text{prime}}}\mathop{\prod }_{i=1}^{d}([-c_{i}r]-[-(c_{i}r+\unicode[STIX]{x1D6FF})]).\end{eqnarray}$$
Let
$$\begin{eqnarray}T(P):=\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\mathop{\prod }_{i=1}^{d}([-c_{i}n]-[-(c_{i}n+\unicode[STIX]{x1D6FF})])\unicode[STIX]{x1D6EC}(n).\end{eqnarray}$$
We aim to show that
$$\begin{eqnarray}T(P)=\unicode[STIX]{x1D6FF}^{d}(b-a\unicode[STIX]{x1D707})P(1+o(1))+O(N^{1-d\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}/2})\quad \text{if }P\unicode[STIX]{x1D707}\leqslant N.\end{eqnarray}$$
As usual, from (15), it follows that
$$\begin{eqnarray}S(P)=\unicode[STIX]{x1D6FF}^{d}(b-a\unicode[STIX]{x1D707})P(\log 2P)^{-1}(1+o(1))+O(N^{1-d\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}/2})\quad \text{if }P\unicode[STIX]{x1D707}\leqslant N,\end{eqnarray}$$
which, together with (13), gives
$$\begin{eqnarray}\displaystyle N(P) & = & \displaystyle R(P)S(P)=\unicode[STIX]{x1D6FF}^{d}(b-a\unicode[STIX]{x1D707})(\unicode[STIX]{x1D707}-1)P^{2}(\log 2P)^{-2}(1+o(1))\nonumber\\ \displaystyle & & \displaystyle +\,O(PN^{1-d\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}/2})\quad \text{if }P\unicode[STIX]{x1D707}\leqslant N.\nonumber\end{eqnarray}$$
Combining this with (8)–(12), we obtain
$$\begin{eqnarray}\displaystyle \displaystyle \mathop{\sum }_{P\leqslant p<\unicode[STIX]{x1D707}P}\unicode[STIX]{x1D706}({\mathcal{B}}_{p}) & {\geqslant} & \displaystyle \frac{(b-a)^{2}}{4a}\cdot \frac{\min (2,c_{1},\ldots ,c_{d})^{d-1}}{2^{d}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,(\unicode[STIX]{x1D707}P)^{-1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}P^{2}(\log 2P)^{-2}(1+o(1))\nonumber\\ \displaystyle & & \displaystyle +\,O(P^{-\unicode[STIX]{x1D6FE}_{d,k}}N^{1-d\unicode[STIX]{x1D6FE}_{d,k}+3\unicode[STIX]{x1D700}/2})\quad \text{if }P\unicode[STIX]{x1D707}\leqslant N.\end{eqnarray}$$
By splitting the interval
$[1,N)$
into intervals of the form
$[P,\unicode[STIX]{x1D707}P)$
and summing up, it now follows, from (7) and (16), that
$$\begin{eqnarray}\displaystyle \int _{a}^{b}F_{N}(\unicode[STIX]{x1D6FC})\,d\unicode[STIX]{x1D6FC} & {\geqslant} & \displaystyle \frac{(b-a)^{2}}{4a}\cdot \frac{\min (2,c_{1},\ldots ,c_{d})^{d-1}}{2^{d}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\biggl(\mathop{\sum }_{k=0}^{\infty }\biggl(\frac{N}{\unicode[STIX]{x1D707}^{k}}\biggr)^{-1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}\biggl(\frac{N}{\unicode[STIX]{x1D707}^{k+1}}\biggr)^{2}(\log N)^{-2}\biggr)(1+o(1))\nonumber\\ \displaystyle & = & \displaystyle \frac{(b-a)^{2}}{4a}\cdot \frac{\min (2,c_{1},\ldots ,c_{d})^{d-1}}{2^{d}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\unicode[STIX]{x1D707}^{-2}\cdot \frac{1}{1-\unicode[STIX]{x1D707}^{-(1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700}))}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,N^{1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}(\log N)^{-2}(1+o(1)).\nonumber\end{eqnarray}$$
Further, since
$\unicode[STIX]{x1D707}>1$
,
$$\begin{eqnarray}\displaystyle 1-\unicode[STIX]{x1D707}^{-(1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700}))} & {\leqslant} & \displaystyle (1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700}))(\unicode[STIX]{x1D707}-1)\nonumber\\ \displaystyle & = & \displaystyle (1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700}))\cdot \frac{b-a}{2a}.\nonumber\end{eqnarray}$$
Hence, we deduce that
$$\begin{eqnarray}\displaystyle \int _{a}^{b}F_{N}(\unicode[STIX]{x1D6FC})\,d\unicode[STIX]{x1D6FC} & {\geqslant} & \displaystyle (b-a)\cdot \frac{2a^{2}}{(a+b)^{2}}\cdot \frac{1}{1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\frac{\min (c_{1},\ldots ,c_{d},2)^{d-1}}{2^{d}}\nonumber\\ \displaystyle & & \displaystyle \times \,N^{1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}(\log N)^{-2}(1+o(1))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle (b-a)\cdot \frac{A^{2}}{B^{2}}\cdot \frac{\min (c_{1},\ldots ,c_{d},2)^{d-1}}{2^{d+1}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,N^{1-(d+1)(\unicode[STIX]{x1D6FE}_{d,k}-\unicode[STIX]{x1D700})}(\log N)^{-2}(1+o(1)),\end{eqnarray}$$
which establishes the claim of Theorem 2.2(i). It remains to prove (15).
3.2 Reduction to exponential sums
For
$x\in \mathbb{R}$
, let
$$\begin{eqnarray}\unicode[STIX]{x1D713}(x):=x-[x]-{\textstyle \frac{1}{2}}.\end{eqnarray}$$
Then we may write
$T(P)$
in the form
$$\begin{eqnarray}\displaystyle T(P) & = & \displaystyle \mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\mathop{\prod }_{i=1}^{d}(\unicode[STIX]{x1D6FF}-(\unicode[STIX]{x1D713}(-c_{i}n)-\unicode[STIX]{x1D713}(-(c_{i}n+\unicode[STIX]{x1D6FF}))))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{{\mathcal{A}}\subseteq \{1,\ldots ,d\}}\unicode[STIX]{x1D6FF}^{d-|{\mathcal{A}}|}T_{{\mathcal{A}}}(P),\end{eqnarray}$$
where
$$\begin{eqnarray}T_{{\mathcal{A}}}(P):=\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\mathop{\prod }_{i\in {\mathcal{A}}}(\unicode[STIX]{x1D713}(-(c_{i}n+\unicode[STIX]{x1D6FF}))-\unicode[STIX]{x1D713}(-c_{i}n)).\end{eqnarray}$$
By the prime number theorem,
$$\begin{eqnarray}T_{\emptyset }(P)=\unicode[STIX]{x1D6FF}^{d}\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\sim \unicode[STIX]{x1D6FF}^{d}(b-a\unicode[STIX]{x1D707})P\quad \text{as }P\rightarrow \infty .\end{eqnarray}$$
Hence, to establish (15), it suffices to prove that, for any fixed
$\unicode[STIX]{x1D700}>0$
, a bound of the form
$$\begin{eqnarray}T_{{\mathcal{A}}}(P)=O(N^{1-d\unicode[STIX]{x1D6FE}_{d,k}+\unicode[STIX]{x1D700}/2})\quad \text{if }P\unicode[STIX]{x1D707}\leqslant N\end{eqnarray}$$
holds for all non-empty subsets
${\mathcal{A}}$
of
$\{1,\ldots ,d\}$
. We reduce the left-hand side to exponential sums, using the following Fourier analytic tool developed by Vaaler [Reference Vaaler19].
Lemma 3.1 (Vaaler).
For
$0<|t|<1$
, let
$$\begin{eqnarray}W(t)=\unicode[STIX]{x1D70B}t(1-|t|)\cot \unicode[STIX]{x1D70B}t+|t|.\end{eqnarray}$$
Fix a positive integer
$J$
. For
$x\in \mathbb{R}$
, define
$$\begin{eqnarray}\unicode[STIX]{x1D713}^{\ast }(x):=-\mathop{\sum }_{1\leqslant |j|\leqslant J}(2\unicode[STIX]{x1D70B}\text{i}j)^{-1}W\biggl(\frac{j}{J+1}\biggr)\text{e}(jx)\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D70F}(x):=\frac{1}{2J+2}\mathop{\sum }_{|j|\leqslant J}\biggl(1-\frac{|j|}{J+1}\biggr)\text{e}(jx).\end{eqnarray}$$
Then
$\unicode[STIX]{x1D70F}(x)$
is non-negative, and
$$\begin{eqnarray}|\unicode[STIX]{x1D713}^{\ast }(x)-\unicode[STIX]{x1D713}(x)|\leqslant \unicode[STIX]{x1D70F}(x)\end{eqnarray}$$
for all real numbers
$x$
.
Proof. This is [Reference Graham and Kolesnik7, Theorem A6] and it has its origin in [Reference Vaaler19]. ◻
We set
$$\begin{eqnarray}\unicode[STIX]{x1D70F}^{\ast }(x):=\unicode[STIX]{x1D713}(x)-\unicode[STIX]{x1D713}^{\ast }(x).\end{eqnarray}$$
Then
$$\begin{eqnarray}\displaystyle T_{{\mathcal{A}}}(P) & = & \displaystyle \mathop{\sum }_{{\mathcal{B}}\subseteq {\mathcal{A}}}\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\biggl(\mathop{\prod }_{i\in {\mathcal{B}}}(\unicode[STIX]{x1D713}^{\ast }(-(c_{i}n+\unicode[STIX]{x1D6FF}))-\unicode[STIX]{x1D713}^{\ast }(-c_{i}n))\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl(\mathop{\prod }_{j\in {\mathcal{A}}\setminus {\mathcal{B}}}(\unicode[STIX]{x1D70F}^{\ast }(-(c_{j}n+\unicode[STIX]{x1D6FF}))-\unicode[STIX]{x1D70F}^{\ast }(-c_{j}n))\biggr)\nonumber\\ \displaystyle & = & \displaystyle U_{{\mathcal{A}}}(P)+O(V_{{\mathcal{A}}}(P)),\end{eqnarray}$$
where
$$\begin{eqnarray}U_{{\mathcal{A}}}(P):=\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\biggl(\mathop{\prod }_{i\in {\mathcal{A}}}(\unicode[STIX]{x1D713}^{\ast }(-(c_{i}n+\unicode[STIX]{x1D6FF}))-\unicode[STIX]{x1D713}^{\ast }(-c_{i}n))\biggr)\end{eqnarray}$$
and
$$\begin{eqnarray}V_{{\mathcal{A}}}(P):=(\log 2P)\mathop{\sum }_{i\in {\mathcal{A}}}\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}(\unicode[STIX]{x1D70F}(-(c_{i}n+\unicode[STIX]{x1D6FF}))+\unicode[STIX]{x1D70F}(-c_{i}n)),\end{eqnarray}$$
where the
$O$
-term
$V_{{\mathcal{A}}}(P)$
arrives by using
$|\unicode[STIX]{x1D713}^{\ast }(x)|\leqslant 2$
,
$|\unicode[STIX]{x1D70F}^{\ast }(x)|\leqslant \unicode[STIX]{x1D70F}(x)\leqslant 1$
,
$\unicode[STIX]{x1D6EC}(n)\leqslant \log n$
and the triangle inequality.
The definition of the function
$\unicode[STIX]{x1D70F}(x)$
gives
$$\begin{eqnarray}V_{{\mathcal{A}}}(P)=\frac{\log 2P}{2J+2}\mathop{\sum }_{i\in {\mathcal{A}}}\mathop{\sum }_{|j|\leqslant J}\biggl(1-\frac{|j|}{J+1}\biggr)(1+\text{e}(j\unicode[STIX]{x1D6FF}))\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\text{e}(jc_{i}n),\end{eqnarray}$$
and the definition of
$\unicode[STIX]{x1D713}^{\ast }(x)$
gives, after multiplying out and rearranging summations,
$$\begin{eqnarray}\displaystyle U_{{\mathcal{A}}}(P) & = & \displaystyle -\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\biggl(\mathop{\prod }_{k=1}^{h}\biggl((2\unicode[STIX]{x1D70B}\text{i}j_{k})^{-1}W\biggl(\frac{j_{k}}{J+1}\biggr)(1+\text{e}(j_{k}\unicode[STIX]{x1D6FF}))\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\text{e}\biggl(n\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr),\nonumber\end{eqnarray}$$
where we suppose that
$J$
is a positive integer satisfying
$J\leqslant P$
and
$$\begin{eqnarray}h:=|{\mathcal{A}}|\quad \text{and}\quad {\mathcal{A}}=\{l_{1},\ldots ,l_{h}\}.\end{eqnarray}$$
We further estimate
$V_{{\mathcal{A}}}(P)$
by
$$\begin{eqnarray}\displaystyle V_{{\mathcal{A}}}(P) & \ll & \displaystyle \frac{P\log 2P}{J}+\frac{\log 2P}{J}\cdot \mathop{\sum }_{i=1}^{d}\mathop{\sum }_{1\leqslant j\leqslant J}\biggl|\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\text{e}(jc_{i}n)\biggr|\nonumber\\ \displaystyle & \ll & \displaystyle \frac{P\log 2P}{J}+\frac{\log 2P}{J}\cdot \mathop{\sum }_{i=1}^{d}\mathop{\sum }_{1\leqslant j\leqslant J}\min (P,\Vert jc_{i}\Vert ^{-1})\nonumber\\ \displaystyle & \ll & \displaystyle \frac{P\log 2P}{J}+(\log 2P)\mathop{\sum }_{i=1}^{d}\mathop{\sum }_{1\leqslant j\leqslant J}\min \biggl(\frac{P}{j},\Vert jc_{i}\Vert ^{-1}\biggr)\nonumber\\ \displaystyle & =: & \displaystyle \frac{P\log 2P}{J}+(\log 2P)\tilde{V}_{d}(P),\end{eqnarray}$$
where the term
$P(\log 2P)/J$
bounds the contribution of
$j=0$
, and we estimate
$U_{{\mathcal{A}}}(P)$
by
$$\begin{eqnarray}\displaystyle & & \displaystyle U_{{\mathcal{A}}}(P)\ll \tilde{U} _{{\mathcal{A}}}(P)\nonumber\\ \displaystyle & & \displaystyle \quad :=\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\biggl|\mathop{\sum }_{Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP}\unicode[STIX]{x1D6EC}(n)\text{e}\biggl(n\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|.\end{eqnarray}$$
It remains to estimate
$\tilde{U} _{{\mathcal{A}}}(P)$
and
$\tilde{V}_{d}(P)$
. The estimation of
$\tilde{V}_{d}(P)$
is clearly easier than that of
$\tilde{U} _{{\mathcal{A}}}(P)$
. We first deal with the term
$\tilde{U} _{{\mathcal{A}}}(P)$
.
3.3 Application of Vaughan’s identity
We convert the inner sum involving the von Mangoldt function on the right-hand side of (22) into bilinear sums using Vaughan’s identity.
Lemma 3.2 (Vaughan).
Let
$u\geqslant 1$
,
$v\geqslant 1$
,
$uv\leqslant x$
. Then, for every arithmetic function
$f:\mathbb{N}\rightarrow \mathbb{C}$
, we have the estimate
$$\begin{eqnarray}\mathop{\sum }_{u<n\leqslant x}f(n)\unicode[STIX]{x1D6EC}(n)\ll (\log 2x)T_{1}+T_{2}\end{eqnarray}$$
with
$$\begin{eqnarray}T_{1}:=\mathop{\sum }_{l\leqslant uv}\max _{w}\biggl|\mathop{\sum }_{w\leqslant m\leqslant x/l}f(ml)\biggr|\end{eqnarray}$$
and
$$\begin{eqnarray}T_{2}:=\biggl|\mathop{\sum }_{u<m\leqslant x/v}\mathop{\sum }_{v<l\leqslant x/m}\unicode[STIX]{x1D6EC}(m)b(l)f(ml)\biggr|,\end{eqnarray}$$
where
$b(l)$
is an arithmetic function that only depends on
$v$
and which satisfies the inequality
$b(l)\leqslant \unicode[STIX]{x1D70F}(l)$
, with
$\unicode[STIX]{x1D70F}(l)$
being the number of divisors of
$l$
.
Proof. This is [Reference Brüdern3, Satz 6.1.2] and it has its origin in [Reference Vaughan20]. ◻
We use Lemma 3.2 with parameters
$u$
and
$v$
satisfying
$1\leqslant u=v\leqslant (Pa\unicode[STIX]{x1D707})^{1/2}$
, to be fixed later,
$x:=bP$
and
$$\begin{eqnarray}f(n):=\left\{\begin{array}{@{}ll@{}}\text{e}(nc)\quad & \text{if }Pa\unicode[STIX]{x1D707}\leqslant n\leqslant bP,\\ 0\quad & \text{if }n<Pa\unicode[STIX]{x1D707},\end{array}\right.\end{eqnarray}$$
with
$$\begin{eqnarray}c:=\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}},\end{eqnarray}$$
to deduce that
$$\begin{eqnarray}\tilde{U} _{{\mathcal{A}}}(P)\ll (\log 2P)Z_{1}+Z_{2},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle Z_{1} & := & \displaystyle \mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\mathop{\sum }_{l\leqslant u^{2}}\max _{Pa\unicode[STIX]{x1D707}/l\leqslant w\leqslant bP/l}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl|\mathop{\sum }_{w\leqslant m\leqslant bP/l}\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|\nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle Z_{2} & := & \displaystyle \mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl|\mathop{\sum }_{u<m\leqslant bP/u}~\mathop{\sum }_{\max (u,Pa\unicode[STIX]{x1D707}/m)\leqslant l\leqslant bP/m}\unicode[STIX]{x1D6EC}(m)b(l)\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|.\nonumber\end{eqnarray}$$
Obviously,
$$\begin{eqnarray}\displaystyle Z_{1}\ll \mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\mathop{\sum }_{l\leqslant u^{2}}\min \biggl(\frac{P}{l},\biggl\|l\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr\|^{-1}\biggr). & & \displaystyle\end{eqnarray}$$
We shall boil down
$Z_{2}$
to similar terms. Rearranging the summation gives
$$\begin{eqnarray}\displaystyle Z_{2} & = & \displaystyle \mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl|\mathop{\sum }_{a\unicode[STIX]{x1D707}u/b\leqslant l\leqslant bP/u}b(l)\mathop{\sum }_{M_{1}(l)\leqslant m\leqslant M_{2}(l)}\unicode[STIX]{x1D6EC}(m)\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|,\nonumber\end{eqnarray}$$
with
$$\begin{eqnarray}M_{1}(l):=\max ([u]+1,Pa\unicode[STIX]{x1D707}/l)\quad \text{and}\quad M_{2}(l):=bP/l.\end{eqnarray}$$
We now observe that
$$\begin{eqnarray}Z_{2}\ll (\log 2P)\max _{u\leqslant L\leqslant bP/u}Z_{2}(L),\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle Z_{2}(L) & := & \displaystyle \mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\mathop{\sum }_{L\leqslant l\leqslant 2L}b(l)\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl|\mathop{\sum }_{M_{1}(l)\leqslant m\leqslant M_{2}(l)}\unicode[STIX]{x1D6EC}(m)\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|.\nonumber\end{eqnarray}$$
Recall that
$1\leqslant J\leqslant P$
. Using the Cauchy–Schwarz inequality and the bound
$$\begin{eqnarray}\mathop{\sum }_{L\leqslant l\leqslant 2L}|b(l)|^{2}\leqslant \mathop{\sum }_{L\leqslant l\leqslant 2L}d(l)^{2}\ll L(\log 2L)^{3},\end{eqnarray}$$
with
$d(l)$
being the divisor function, and by expanding the square, we obtain
$$\begin{eqnarray}\displaystyle Z_{2}(L)^{2} & \ll & \displaystyle \biggl(\mathop{\sum }_{L\leqslant l\leqslant 2L}|b(l)|^{2}\biggr)\biggl(\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{L\leqslant l\leqslant 2L}\biggl|\mathop{\sum }_{M_{1}(l)\leqslant m\leqslant M_{2}(l)}\unicode[STIX]{x1D6EC}(m)\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|^{2}\nonumber\\ \displaystyle & \ll & \displaystyle \biggl(\mathop{\sum }_{L\leqslant l\leqslant 2L}|b(l)|^{2}\biggr)(\log 2P)^{h}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{L\leqslant l\leqslant 2L}\biggl|\mathop{\sum }_{M_{1}(l)\leqslant m\leqslant M_{2}(l)}\unicode[STIX]{x1D6EC}(m)\text{e}\biggl(ml\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\biggr|^{2}.\nonumber\end{eqnarray}$$
Expanding the square, and splitting the resulting expression into a diagonal and non-diagonal term, we estimate the above further by
$$\begin{eqnarray}\displaystyle Z_{2}(L)^{2} & \ll & \displaystyle L(\log 2P)^{h+3}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{L\leqslant l\leqslant 2L}\mathop{\sum }_{M_{1}(l)\leqslant m\leqslant M_{2}(l)}\unicode[STIX]{x1D6EC}(m)^{2}\nonumber\\ \displaystyle & & \displaystyle +\,L(\log 2P)^{h+3}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl|\mathop{\sum }_{L\leqslant l\leqslant 2L}~\mathop{\sum }_{M_{1}(l)\leqslant m_{1}<m_{2}\leqslant M_{2}(l)}\!\!\unicode[STIX]{x1D6EC}(m_{1})\unicode[STIX]{x1D6EC}(m_{2})\text{e}\biggl(\!(m_{2}-m_{1})l\!\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\!\biggr)\!\biggr|.\nonumber\end{eqnarray}$$
Exchanging summation and estimating geometric sums, we deduce that
$$\begin{eqnarray}\displaystyle Z_{2}(L)^{2} & \ll & \displaystyle LP(\log 2P)^{2h+5}+L(\log 2P)^{h+3}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\sum }_{u\leqslant m_{1}<m_{2}\leqslant bP/L}\bigg|\mathop{\sum }_{\max (L,Pa\unicode[STIX]{x1D707}/m_{1})\leqslant l\leqslant \min (2L,bP/m_{2})}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \times \,\text{e}\biggl((m_{2}-m_{1})l\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr)\bigg|\nonumber\\ \displaystyle & \ll & \displaystyle LP(\log 2P)^{2h+5}+L(\log 2P)^{h+3}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{u\leqslant m_{1}<m_{2}\leqslant bP/L}\min \biggl(L,\biggl|\biggl|(m_{2}-m_{1})\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr|\biggr|^{-1}\biggr)\nonumber\\ \displaystyle & \ll & \displaystyle LP(\log 2P)^{2h+5}+P(\log 2P)^{h+3}\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\sum }_{1\leqslant m\leqslant bP/L}\min \biggl(\frac{P}{m},\biggl|\biggl|m\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr|\biggr|^{-1}\biggr).\end{eqnarray}$$
3.4 Construction of a Dirichlet approximation
The treatments of
$Z_{1}$
and
$Z_{2}$
lead to sums of the form
$$\begin{eqnarray}R_{{\mathcal{A}}}(M,x):=\mathop{\sum }_{1\leqslant |j_{1}|\leqslant J}\!\cdots \mathop{\sum }_{1\leqslant |j_{h}|\leqslant J}\frac{1}{|j_{1}\cdots j_{h}|}\mathop{\sum }_{1\leqslant m\leqslant M}\!\min \biggl(\frac{x}{m},\biggl\|m\mathop{\sum }_{k\in {\mathcal{A}}}j_{k}c_{l_{k}}\biggr\|^{-1}\biggr),\end{eqnarray}$$
which we estimate in the following. Here the Diophantine properties of the vector
$(c_{1},\ldots ,c_{d})$
come into play.
Splitting each of the
$j_{k}$
-intervals into
$O(\log P)$
dyadic intervals and summing up their contributions, we obtain
$$\begin{eqnarray}R_{{\mathcal{A}}}(M,x)\ll (\log 2P)^{d}\sup _{1\leqslant H_{1},\ldots ,H_{d}\leqslant J}\frac{R_{d}(H_{1},\ldots ,H_{d},M,x)}{H_{1}\cdots H_{d}},\end{eqnarray}$$
where
$$\begin{eqnarray}R_{d}(H_{1},\ldots ,H_{d},M,x):=\mathop{\sum }_{\substack{ |j_{1}|\leqslant H_{1},\ldots ,|j_{d}|\leqslant H_{d} \\ \mathbf{j}\not =\mathbf{0}}}\mathop{\sum }_{1\leqslant m\leqslant M}\min \biggl(\frac{x}{m},\Vert m\mathbf{j}\cdot \mathbf{c}\Vert ^{-1}\biggr),\end{eqnarray}$$
with
$$\begin{eqnarray}\mathbf{j}:=(j_{1},\ldots ,j_{d})\quad \text{and}\quad \mathbf{c}:=(c_{1},\ldots ,c_{d}).\end{eqnarray}$$
The key point is now to approximate
$\mathbf{j}\,\cdot \,\mathbf{c}$
by rational numbers, using Dirichlet’s approximation theorem, where the denominators are uniformly bounded from below. Let
$\mathbf{j}\in \mathbb{Z}^{d}\setminus \mathbf{0}$
and let
$X$
be a positive integer, to be fixed later. By Dirichlet’s approximation theorem, there exist integers
$a,q$
such that gcd
$(a,q)=1$
,
$1\leqslant q\leqslant X$
and
$$\begin{eqnarray}\biggl|\mathbf{j}\cdot \mathbf{c}-\frac{a}{q}\biggr|\leqslant \frac{1}{qX}.\end{eqnarray}$$
Hence,
$$\begin{eqnarray}\Vert q\mathbf{j}\cdot \mathbf{c}\Vert \leqslant \frac{1}{X}.\end{eqnarray}$$
On the other hand, by condition (1) in Theorem 1.1,
$$\begin{eqnarray}\frac{C}{q^{k}\Vert \mathbf{j}\Vert _{\infty }^{k}}<\Vert q\mathbf{j}\cdot \mathbf{c}\Vert .\end{eqnarray}$$
It follows that
$$\begin{eqnarray}q>\frac{(CX)^{1/k}}{\Vert \mathbf{j}\Vert _{\infty }}.\end{eqnarray}$$
Now we apply the following well-known lemma.
Lemma 3.3. Let
$L\geqslant 1$
and
$x>1$
. Suppose that
$|c-a/q|\leqslant q^{-2}$
with
$a\in \mathbb{Z}$
,
$q\in \mathbb{N}$
and
$(a,q)=1$
. Then
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant l\leqslant L}\min \biggl(\frac{x}{l},\Vert lc\Vert ^{-1}\biggr)\ll \biggl(\frac{x}{q}+L+q\biggr)(\log 2Lqx).\end{eqnarray}$$
Proof. This is Lemma 6.4.4 in [Reference Brüdern3]. ◻
It follows that
$$\begin{eqnarray}\mathop{\sum }_{1\leqslant m\leqslant M}\min \biggl(\frac{x}{m},\Vert m\mathbf{j}\cdot \mathbf{c}\Vert ^{-1}\biggr)\ll \biggl(\frac{x\Vert \mathbf{j}\Vert _{\infty }}{X^{1/k}}+M+X\biggr)(\log 2MXx)\end{eqnarray}$$
and hence
$$\begin{eqnarray}\displaystyle & & \displaystyle R_{d}(H_{1},\ldots ,H_{d},M,x)\nonumber\\ \displaystyle & & \displaystyle \quad \ll H_{1}\cdots H_{d}\biggl(\frac{x\max (H_{1},\ldots ,H_{d})}{X^{1/k}}+M+X\biggr)(\log 2MXx).\end{eqnarray}$$
Combining (28) and (30), we obtain
$$\begin{eqnarray}R_{{\mathcal{A}}}(M,x)\ll (\log 2P)^{d+1}\biggl(\frac{xJ}{X^{1/k}}+M+X\biggr).\end{eqnarray}$$
Now, choosing
$$\begin{eqnarray}X:=[(xJ)^{1-1/(k+1)}],\end{eqnarray}$$
we deduce that
$$\begin{eqnarray}R_{{\mathcal{A}}}(M,x)\ll (\log 2P)^{d+1}(M+(xJ)^{1-1/(k+1)}).\end{eqnarray}$$
3.5 Completion of the proof
Using (24), (26), (28) and (31), we get
$$\begin{eqnarray}Z_{1}\ll (\log 2P)^{d+1}(u^{2}+(PJ)^{1-1/(k+1)})\end{eqnarray}$$
and
$$\begin{eqnarray}Z_{2}(L)\ll (\log 2P)^{3d/2+5/2}(P^{1/2}L^{1/2}+PL^{-1/2}+P^{1-1/(2(k+1))}J^{1/2-1/(2(k+1))})\end{eqnarray}$$
and hence, by (25),
$$\begin{eqnarray}Z_{2}\ll (\log 2P)^{3d/2+7/2}(Pu^{-1/2}+P^{1-1/(2(k+1))}J^{1/2-1/(2(k+1))}).\end{eqnarray}$$
Combining (22), (23), (32) and (33), we get
$$\begin{eqnarray}\tilde{U} _{{\mathcal{A}}}(P)\ll P^{\unicode[STIX]{x1D700}}(u^{2}+Pu^{-1/2}+(PJ)^{1-1/(k+1)}+P^{1-1/(2(k+1))}J^{1/2-1/(2(k+1))}).\end{eqnarray}$$
Now, choosing
$u:=P^{2/5}$
, it follows that
$$\begin{eqnarray}\tilde{U} _{{\mathcal{A}}}(P)\ll P^{\unicode[STIX]{x1D700}}(P^{4/5}+(PJ)^{1-1/(k+1)}+P^{1-1/(2(k+1))}J^{1/2-1/(2(k+1))}).\end{eqnarray}$$
We now turn to the term
$\tilde{V}_{d}(P)$
, defined in (21). By (29) and (30),
$$\begin{eqnarray}\tilde{V}_{d}(P)\ll R_{d}(1,\ldots ,1,J,P)\ll \biggl(\frac{P}{X^{1/k}}+J+X\biggr)(\log 2JXP)\end{eqnarray}$$
for any positive integer
$X$
. Choosing
$$\begin{eqnarray}X:=[P^{1-1/(k+1)}]\end{eqnarray}$$
gives
$$\begin{eqnarray}\tilde{V}_{d}(P)\ll \log (2P)(J+P^{1-1/(k+1)}).\end{eqnarray}$$
Combining this with (21), we find that
$$\begin{eqnarray}V_{{\mathcal{A}}}(P)\ll (\log 2P)(PJ^{-1}+J+P^{1-1/(k+1)}).\end{eqnarray}$$
Now, putting (20), (34) and (35) together, we arrive at
$$\begin{eqnarray}T_{{\mathcal{A}}}(P)\ll P^{\unicode[STIX]{x1D700}}(PJ^{-1}+J+P^{4/5}+(PJ)^{1-1/(k+1)}+P^{1-1/(2(k+1))}J^{1/2-1/(2(k+1))}).\end{eqnarray}$$
Now, choosing
$$\begin{eqnarray}J:=[P^{1/(3k+2)}],\end{eqnarray}$$
we get
$$\begin{eqnarray}T_{{\mathcal{A}}}(P)\ll P^{1-1/(3k+2)+\unicode[STIX]{x1D700}},\end{eqnarray}$$
which proves (19), upon replacing
$\unicode[STIX]{x1D700}$
by
$\unicode[STIX]{x1D700}/2$
. This completes the proof of Theorem 2.2(i).
4 Proof of Theorem 2.2(ii)
4.1 Sieve theoretical approach
We are broadly following the treatment in [Reference Harman and Jones12] with appropriate modifications because the linear case, considered here, requires a different treatment. In particular, as in the previous section, the Diophantine properties of the vector
$(c_{1},\ldots ,c_{d})$
will come into play. Let
$\{\cdot \}$
represent the fractional part, and put
$$\begin{eqnarray}\unicode[STIX]{x1D707}:=N^{\unicode[STIX]{x1D700}-\unicode[STIX]{x1D6FE}_{d,k}}.\end{eqnarray}$$
Write
$$\begin{eqnarray}{\mathcal{A}}={\mathcal{A}}(\unicode[STIX]{x1D6FC})=\{n[n\unicode[STIX]{x1D6FC}]:1\leqslant n\leqslant N,\{n\unicode[STIX]{x1D6FC}\}<\unicode[STIX]{x1D707},\{nc_{i}\unicode[STIX]{x1D6FC}\}<\unicode[STIX]{x1D707}\text{ for }i=1,\ldots ,d\}.\end{eqnarray}$$
We desire to show that
${\mathcal{A}}$
does not contain too many products of two primes. To this end, we apply a two-dimensional upper bound sieve (see [Reference Halberstam and Richert8], Theorem 5.2). We therefore need to obtain an asymptotic formula for the number of solutions to
$$\begin{eqnarray}n[n\unicode[STIX]{x1D6FC}]\equiv 0\hspace{0.2em}\bmod \hspace{0.2em}q,\quad 1\leqslant n\leqslant N,\end{eqnarray}$$
with
$$\begin{eqnarray}\{n\unicode[STIX]{x1D6FC}\}<\unicode[STIX]{x1D707}\quad \text{and}\quad \{nc_{i}\unicode[STIX]{x1D6FC}\}<\unicode[STIX]{x1D707}\quad \text{for }i=1,\ldots ,d,\end{eqnarray}$$
where
$$\begin{eqnarray}q\leqslant Q:=N^{\unicode[STIX]{x1D700}}.\end{eqnarray}$$
For this, it suffices to establish a formula for the number of solutions to
$$\begin{eqnarray}n\equiv 0\hspace{0.2em}\bmod \hspace{0.2em}t_{1},\qquad [n\unicode[STIX]{x1D6FC}]\equiv 0\hspace{0.2em}\bmod \hspace{0.2em}t_{2},\end{eqnarray}$$
subject to (37). We can combine (37) with the congruence conditions to require
$$\begin{eqnarray}1\leqslant n\leqslant \frac{N}{t_{1}},\qquad \biggl\{\frac{nt_{1}\unicode[STIX]{x1D6FC}}{t_{2}}\biggr\}<\frac{\unicode[STIX]{x1D707}}{t_{2}},\qquad \{nt_{1}c_{i}\unicode[STIX]{x1D6FC}\}<\unicode[STIX]{x1D707}\quad \text{for }i=1,\ldots ,d,\end{eqnarray}$$
and count the number
$S(\unicode[STIX]{x1D6FC};t_{1},t_{2})$
of solutions to (38), using Fourier analysis. To this end, we write
$$\begin{eqnarray}S(\unicode[STIX]{x1D6FC},t_{1},t_{2})=\mathop{\sum }_{1\leqslant n\leqslant N/t_{1}}\biggl(\biggl[\frac{nt_{1}\unicode[STIX]{x1D6FC}}{t_{2}}\biggr]-\biggl[\frac{nt_{1}\unicode[STIX]{x1D6FC}}{t_{2}}-\frac{\unicode[STIX]{x1D707}}{t_{2}}\biggr]\biggr)\cdot \mathop{\prod }_{i=1}^{d}([nt_{1}c_{i}\unicode[STIX]{x1D6FC}]-[nt_{1}c_{i}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D707}])\end{eqnarray}$$
and evaluate this term in a similar way as the term
$T(P)$
defined in (14), using Vaaler’s Lemma 3.1. By a chain of similar calculations, we arrive at the asymptotic estimate
$$\begin{eqnarray}S(\unicode[STIX]{x1D6FC};t_{1},t_{2})=\frac{N\unicode[STIX]{x1D707}^{d+1}}{t_{1}t_{2}}+O\biggl(\frac{N\unicode[STIX]{x1D707}^{d}}{L}+E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\biggr),\end{eqnarray}$$
where we set
$$\begin{eqnarray}L:=Q^{3}\unicode[STIX]{x1D707}^{-1}\end{eqnarray}$$
and
$$\begin{eqnarray}E(\unicode[STIX]{x1D6FC};t_{1},t_{2}):=\frac{\unicode[STIX]{x1D707}^{d+1}}{t_{2}}\mathop{\sum }_{\substack{ |m_{0}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{0},\ldots ,m_{d})\not =(0,\ldots ,0)}}\biggl|\mathop{\sum }_{1\leqslant n\leqslant N/t_{1}}\text{e}\biggl(n\unicode[STIX]{x1D6FC}t_{1}\biggl(\frac{m_{0}}{t_{2}}+\mathop{\sum }_{i=1}^{d}c_{i}m_{i}\biggr)\biggr)\biggr|.\end{eqnarray}$$
The above estimate (39) is analogous to the equation after (26) in [Reference Harman and Jones12].
Now, applying the upper bound sieve gives
$$\begin{eqnarray}F_{N}(\unicode[STIX]{x1D6FC})\leqslant \frac{C(\unicode[STIX]{x1D700})N\unicode[STIX]{x1D707}^{d}}{\log ^{2}N}+O(J_{N}(\unicode[STIX]{x1D6FC})),\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle J_{N}(\unicode[STIX]{x1D6FC}) & := & \displaystyle \mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}\biggl(\frac{N\unicode[STIX]{x1D707}^{d}}{L}+E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})+o\biggl(\frac{N\unicode[STIX]{x1D707}^{d+1}}{\log ^{2}N}\biggr)\quad \text{as }N\rightarrow \infty .\nonumber\end{eqnarray}$$
Hence, to establish the claim in Theorem 2.2(i), it suffices to show that
$$\begin{eqnarray}\mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}\int _{A}^{B}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}=o\biggl(\frac{N\unicode[STIX]{x1D707}^{d+1}}{\log ^{2}N}\biggr)\quad \text{as }N\rightarrow \infty ,N\in {\mathcal{S}}.\end{eqnarray}$$
4.2 Average estimation for
$E(\unicode[STIX]{x1D6FC};t_{1},t_{2})$
To estimate the expression on the right-hand side of (41), we first observe that
$$\begin{eqnarray}\displaystyle & & \displaystyle E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\nonumber\\ \displaystyle & & \displaystyle \quad \ll \frac{\unicode[STIX]{x1D707}^{d+1}}{t_{2}}\mathop{\sum }_{\substack{ |m_{0}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{0},\ldots ,m_{d})\not =(0,\ldots ,0)}}\min \biggl(\frac{N}{t_{1}},\biggl\|\unicode[STIX]{x1D6FC}t_{1}\biggl(\frac{m_{0}}{t_{2}}+m_{1}c_{1}+\cdots +m_{d}c_{d}\biggr)\biggr\|^{-1}\biggr)\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
and note that if
$(m_{0},\ldots ,m_{d})\not =(0,\ldots ,0)$
, then the term
$$\begin{eqnarray}t_{1}\biggl(\frac{m_{0}}{t_{2}}+m_{1}c_{1}+\cdots +m_{d}c_{d}\biggr)\end{eqnarray}$$
on the right-hand side of (42) is non-zero because
$(c_{1},\ldots ,c_{d})$
is
$k$
-Diophantine. For an estimation of the integral in (41), we now use the following lemma.
Lemma 4.1. Suppose that
$0<A<B$
,
$K\geqslant 2$
and
$x\not =0$
. Then
$$\begin{eqnarray}\int _{A}^{B}\min (K,\Vert \unicode[STIX]{x1D6FC}x\Vert ^{-1})\,d\unicode[STIX]{x1D6FC}=O_{A,B}(\min \{K,\max \{1,|x|^{-1}\}\}\log K).\end{eqnarray}$$
Proof. We confine ourselves to the case when
$x>0$
since the case when
$x<0$
is similar. By change of variables
$\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}x$
, we get
$$\begin{eqnarray}\int _{A}^{B}\min (K,\Vert \unicode[STIX]{x1D6FC}x\Vert ^{-1})\,d\unicode[STIX]{x1D6FC}=\frac{1}{x}\int _{xA}^{xB}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}.\end{eqnarray}$$
By periodicity of the integrand, if
$x(B-A)\geqslant 1$
, then
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{x}\int _{xA}^{xB}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{[x(B-A)+1]}{x}\int _{0}^{1}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}=O_{A,B}(\log K).\end{eqnarray}$$
If
$1/K\leqslant x(B-A)<1$
, then
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{x}\int _{xA}^{xB}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \frac{1}{x}\int _{-x(B-A)/2}^{x(B-A)/2}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}=O_{A,B}\biggl(\frac{\log K}{x}\biggr).\end{eqnarray}$$
If
$0<x(B-A)<1/K$
, then, trivially,
$$\begin{eqnarray}\frac{1}{x}\int _{xA}^{xB}\min (K,\Vert \unicode[STIX]{x1D6FD}\Vert ^{-1})\,d\unicode[STIX]{x1D6FD}=O_{A,B}(K).\end{eqnarray}$$
Combining (44)–(47), we deduce the claim when
$x>0$
, which completes the proof.◻
Now, employing (42) and Lemma 4.1, we get
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{A}^{B}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}\ll \frac{\unicode[STIX]{x1D707}^{d+1}}{t_{2}}\mathop{\sum }_{\substack{ |m_{0}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{0},\ldots ,m_{d})\not =(0,\ldots ,0)}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\min \bigg\{\frac{N}{t_{1}},\max \biggl\{1,\biggl|t_{1}\biggl(\frac{m_{0}}{t_{2}}+m_{1}c_{1}+\cdots +m_{d}c_{d}\biggr)\biggr|^{-1}\biggr\}\log N\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad \ll \unicode[STIX]{x1D707}^{d+1}(\log N)\mathop{\sum }_{\substack{ |m_{0}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{0},\ldots ,m_{d})\not =(0,\ldots ,0)}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\min \biggl\{\frac{N}{t_{1}t_{2}},\max \biggl\{\frac{1}{t_{2}},|t_{1}m_{0}+t_{1}t_{2}(m_{1}c_{1}+\cdots +m_{d}c_{d})|^{-1}\biggr\}\biggr\}.\nonumber\end{eqnarray}$$
The contribution of
$(m_{1},\ldots ,m_{d})=(0,\ldots ,0)$
to the last line is bounded by
$O(\unicode[STIX]{x1D707}^{d+1}L\log N)$
, and the contribution of
$(m_{1},\ldots ,m_{d})\not =0$
is bounded by
$$\begin{eqnarray}\displaystyle & & \displaystyle \quad \ll \unicode[STIX]{x1D707}^{d+1}(\log N)\mathop{\sum }_{\substack{ |m_{1}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{1},\ldots ,m_{d})\not =(0,\ldots ,0)}}\mathop{\sum }_{|m_{0}|\leqslant L}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\min \biggl\{\frac{N}{t_{1}t_{2}},\max \biggl\{\frac{1}{t_{2}},|t_{1}m_{0}+t_{1}t_{2}(m_{1}c_{1}+\cdots +m_{d}c_{d})|^{-1}\biggr\}\biggr\}\nonumber\\ \displaystyle & & \displaystyle \quad \ll \unicode[STIX]{x1D707}^{d+1}(\log N)\mathop{\sum }_{\substack{ |m_{1}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{1},\ldots ,m_{d})\not =(0,\ldots ,0)}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\min \biggl\{\frac{N}{t_{1}t_{2}},\Vert t_{1}t_{2}(m_{1}c_{1}+\cdots +m_{d}c_{d})\Vert ^{-1}\biggr\}+\unicode[STIX]{x1D707}^{d+1}L^{d+1}\log N.\nonumber\end{eqnarray}$$
So, altogether,
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{A}^{B}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}\ll \unicode[STIX]{x1D707}^{d+1}L^{d+1}\log N\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D707}^{d+1}(\log N)\mathop{\sum }_{\substack{ |m_{1}|\leqslant L,\ldots ,|m_{d}|\leqslant L \\ (m_{1},\ldots ,m_{d})\not =(0,\ldots ,0)}}\min \biggl\{\frac{N}{t_{1}t_{2}},\Vert t_{1}t_{2}(m_{1}c_{1}+\cdots +m_{d}c_{d})\Vert ^{-1}\biggr\}.\nonumber\end{eqnarray}$$
Summing up over
$t_{1}$
and
$t_{2}$
, writing
$q=t_{1}t_{2}$
and using
$d(q)=O(q^{\unicode[STIX]{x1D700}})$
, with
$d(q)$
being the divisor function, we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}\int _{A}^{B}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}\nonumber\\ \displaystyle & & \displaystyle \quad \ll N^{\unicode[STIX]{x1D700}}QL^{d+1}\unicode[STIX]{x1D707}^{d+1}+N^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D707}^{d+1}R_{d}(L,\ldots ,L,Q,N),\nonumber\end{eqnarray}$$
where
$R_{d}(L,\ldots ,L,Q,N)$
is defined as in (29). Combining this with (30), we get
$$\begin{eqnarray}\mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}\!\int _{A}^{B}\!E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}\ll N^{\unicode[STIX]{x1D700}}L^{d}\unicode[STIX]{x1D707}^{d+1}\biggl(L+\frac{LN}{X^{1/k}}+Q+X\biggr)\log (2QNX)\end{eqnarray}$$
for any positive integer
$X$
. Choosing
$X:=[(LN)^{1-1/(k+1)}]$
, we deduce that
$$\begin{eqnarray}\mathop{\sum }_{t_{1}t_{2}\leqslant Q}(t_{1}t_{2})^{\unicode[STIX]{x1D700}}\int _{A}^{B}E(\unicode[STIX]{x1D6FC};t_{1},t_{2})\,d\unicode[STIX]{x1D6FC}\ll N^{\unicode[STIX]{x1D700}}L^{d}\unicode[STIX]{x1D707}^{d+1}(L+Q+(LN)^{1-1/(k+1)}),\end{eqnarray}$$
which implies (41), upon recalling that
$\unicode[STIX]{x1D707}:=N^{\unicode[STIX]{x1D700}-\unicode[STIX]{x1D6FE}_{d,k}}$
,
$Q:=N^{\unicode[STIX]{x1D700}}$
,
$L:=Q^{3}\unicode[STIX]{x1D707}^{-1}$
and (2).
Acknowledgements
S. Baier wishes to thank the Tata Institute of Fundamental Research in Mumbai (India) for its warm hospitality and excellent working conditions. This work was completed while Ghosh was a visiting professor at the Technion, Israel Institute of Technology and a member at MSRI Berkeley. The hospitality of both institutions is gratefully acknowledged.
