On the exponential large sieve inequality for sparse sequences modulo primes

TL;DR

This paper improves the large sieve inequality for sparse exponential sequences modulo primes, extending its applicability to more rapidly increasing sequences and applying it to digit-prescribed integers.

Contribution

It introduces a stronger version of the large sieve inequality for sparse exponential sequences, surpassing previous bounds and enabling new applications.

Findings

Enhanced inequality valid for sequences with s_n ≤ n^{2+o(1)}

Applicable to sequences with faster growth than previous bounds

Applied to arithmetic properties of integers with prescribed digits

Abstract

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $\lambda^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.

Equations

$${\mathcal{E}}_{\Delta }(X)=\{p\le X:t_p\ge \Delta \}.$$

$$V_{\lambda }(\Gamma ,\mathcal{S};T,X,\Delta )=\sum _{p\in {\mathcal{E}}_{\Delta }(X)}\underset{\gcd (a,p)=1}{\max }{\left|\sum _{n\le T}{\gamma }_n{\mathbf{e}}_p(a{\lambda }^{s_n})\right|}^2,$$

$$S\le X^{15/14+o(1)}.$$

$$V_{\lambda }(\Gamma ,\mathcal{S};T,X)=\sum _{p\in X}\underset{\gcd (a,p)=1}{\max }{\left|\sum _{n\le T}{\gamma }_n{\mathbf{e}}_p(a{\lambda }^{s_n})\right|}^2.$$

$$\underset{(a,p)=1}{\max }\left|\sum _{z=1}^t{e}_p(a{\lambda }^z)\right|\le t^{\alpha }{p}^{\beta +o(1)},$$

$$(\alpha ,\beta )=(0,1/2),$$

$$(\alpha ,\beta )=(5/8,1/8),$$

$$(\alpha ,\beta )=(3/8,1/4),$$

$$(\alpha ,\beta )=(1/2,1/6),$$

$$(1-\vartheta ,\zeta \vartheta ),$$

$$\frac{\beta +\eta }{1-\alpha }\le \frac{1}{2}-\delta .$$

$$T^{1+1/(3-2\alpha )}\ge S{X}^{2\eta }.$$

$$X\le {\left({\left(\frac{T}{S{X}^{2\eta }}\right)}^{1/(3-2\alpha )}\Delta \right)}^k.$$

$$V_{\lambda }(\Gamma ,\mathcal{S}$$

$$\le \left(X+T{X}^{-\delta /(k^2+2)}+{\left(S^{2-2\alpha }T{X}^{-2\eta }\right)}^{1/(3-2\alpha )}\right)T{X}^{1+o(1)}.$$

$$X\le {\left(T^{-1/(3-2\alpha {)}^2}\Delta \right)}^k.$$

$$V_{\lambda }(\Gamma ,\mathcal{S};T,X)\le \left(X^{1-\rho }{T}^2+X^{3/2}{T}^{3/2}+X^{3/4}{T}^{7/8}{S}^{1/4}\right)X^{o(1)}.$$

$$T>X^{1+\epsilon }\text{and}S<T{X}^{1+\epsilon },$$

$$D=\underset{0\le a\le b\le 1}{\sup }\left|\frac{A(a,b)}{T}-(b-a)\right|,$$

$${\mathcal{E}}_{X^{1/2},X}=(1+o(1))\frac{X}{\log X},X\to \infty ,$$

$$A(\lambda ,p)={\left\{\frac{{\lambda }^{s_n}}{p}\mathrm{mod}1\right\}}_{1\le n\le T},$$

$$D\le (\log T{)}^{-\delta },$$

$$D\le T^{-\delta },$$

$$S\le T^{2-\epsilon },$$

$$\omega \left(\prod _{z\in \mathcal{N}(a;\mathcal{S})}z\right)\gg T^{1+\delta }$$

$$\text{sideset}{}^{\ast }{\sum }_{c\mathrm{mod}k}\psi (c)=\sum _{\begin{array}{c}c=1\\ \gcd (c,k)=1\end{array}}^k\psi (c).$$

$$\sum _{k\le K}\text{sideset}{}^{\ast }{\sum }_{c\mathrm{mod}k}{\left|\sum _{n\le T}{\gamma }_n{\mathbf{e}}_k(c{s}_n)\right|}^2\ll (K^2+S)T.$$

$$\underset{(a,\ell )=1}{\max }\left|\sum _{x=0}^{t-1}{\mathbf{e}}_{\ell }(a{g}_{t,\ell }^x)\right|\ll t{\ell }^{1/2k^2}(t^{-1/k}+U^{-1/k^2}),$$

$$\#\{p\text{prime}:{\mathrm{ord}}_p\lambda \le Z\}\ll Z^2.$$

$$\#\{p\text{prime}:{\mathrm{ord}}_p\lambda <Z\}\le \omega \left(\prod _{1\le z\le Z}\left({\lambda }^z-1\right)\right),$$

$$\#\{p\text{prime}:{\mathrm{ord}}_p\lambda <Z\}\ll \log \prod _{1\le z\le Z}({\lambda }^z-1)\le \log \left({\lambda }^{Z^2/2}\right)\ll Z^2,$$

Full text

1. Introduction

The classical large sieve inequality, giving upper bounds on average values of various trigonometric and Dirichlet polynomials with essentially arbitrary sequences ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ , has proved to be an extremely useful and versatile tool in analytic number theory and harmonic analysis, see, for example, [13, 17, 18]. Garaev and Shparlinski [10, Theorem 3.1] have introduced a modification of the large sieve, for both trigonometric and Dirichlet polynomials with arguments that contain exponentials of ${\mathcal{S}}$ rather than the elements of ${\mathcal{S}}$. In the case of trigonometric polynomials, Garaev [9] has introduced a new approach, which has led to a stronger version of the the exponential large sieve inequality, improving some of the results of [10], see also [1, Lemma 2.11] and [22, Theorem 1] for several other bounds of this type. Furthermore, stronger versions of the exponential large sieve inequality for special sequences ${\mathcal{S}}$, such as $T$ consecutive integers or the first $T$ primes, can also be found in [1, 10], with some applications given in [21].

Here we continue this direction and concentrate on the case of general sequences ${\mathcal{S}}$ without any arithmetic restriction. We introduce several new ideas which allow us to improve some results of Garaev [9]. For example, we make use of the bound of [15, Theorem 5.5] on exponential sums over small multiplicative subgroups modulo $p$, which hold for almost all primes $p$, see Lemma 3.2. We also make the method more flexible so it now applies to much sparser sequences ${\mathcal{S}}$ than in [9].

More precisely, let us fix some integer $\lambda\geq 2$. For each prime number $p$, we let $t_{p}$ denote the order of $\lambda\bmod{p}$. For real $X$ and $\Delta$ we define the set

[TABLE]

Note that by a result of Erdös and Murty [8], see also (2.8), for $\Delta=X^{1/2}$ almost all primes $p\leq X$ belong to ${\mathcal{E}}_{\Delta}(X)$.

For integer $T$ and two sequences of complex weights $\Gamma=\{\gamma_{n}\}_{n=1}^{T}$ and integers ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ we define the sums

[TABLE]

where ${\mathbf{\,e}}_{r}(z)=\exp(2\pi iz/r)$.

These sums majorize the ones considered by Garaev [9] where each term is divided by the divisor function $\tau(p-1)$ of $p-1$ . Here we obtain a new bound of the sums $V_{\lambda}(\Gamma,{\mathcal{S}};T,X,\Delta)$ which in particular improves some bounds of Garaev [9].

The argument of Garaev [9] reduces the problem to bounding Gauss sums for which he uses the bound of Heath-Brown and Konyagin [12], that is, the admissible pair (2.1), which is defined below. In particular, for $V_{\lambda}(\Gamma,{\mathcal{S}};T,X,X^{1/2})$ the result of Garaev [9] is nontrivial provided

[TABLE]

Our results by-pass significantly the threshold (1.1) allow to replace $15/14$ with any fixed $\vartheta<2$.

Our improvement is based on a modification of the argument of Garaev [9] which allows us to use the bounds of short sums with exponential functions, given in [15, Theorem 5.5], see also Lemma 3.2 below. This alone allows us to extend the result of [9] to sparse sequences ${\mathcal{S}}$, roughly growing as at most $s_{n}\leq n^{7/6-\varepsilon}$ for any fixed $\varepsilon>0$ in the same scenario where the result of [9] limits the growth to $s_{n}\leq n^{15/14-\varepsilon}$. Furthermore, using bounds of exponential sums over small subgroups of finite fields, in particular that of Bourgain, Glibichuk and Konyagin [5] we relax the condition on ${\mathcal{S}}$ to $s_{n}\leq n^{3/2-\varepsilon}$.

Using a different argument which combines a bound of Bourgain and Chang [4] for Gauss sums modulo a product of two primes with a duality principle for bilinear forms, we obtain another, although less explicit bound which allows the elements to grow as fast as $s_{n}\leq n^{2-\varepsilon}$. Furthermore, for this result we do not need to limit the summation to primes from ${\mathcal{E}}_{\Delta}(X)$ but can consider all primes from $p\leq X$, in which case we denote

[TABLE]

We also give an application of our new estimate to investigating arithmetic properties of integers with almost all digits prescribed in some fixed base. To simplify the exposition, we only consider binary expansions (and hence we talk about bits rather than binary digits). Namely, for an integer $S\geq 1$, an $S$-bit integer $a$ and a sequence of integers ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ with $0\leq s_{1}<\ldots<s_{T}\leq S$, we denote by ${\mathcal{N}}(a;{\mathcal{S}})$ the set of $S$-bit integers $z$ whose bits on all positions $j=1,\ldots,S$ (counted from the right) must agree with those of $a$ except maybe when $j\in{\mathcal{S}}$.

We first recall that Bourgain [2, 3] has recently obtained several very strong results about the distribution of prime numbers among the elements of ${\mathcal{N}}(a;{\mathcal{S}})$, see also [11]. However, in the setting of the strongest result in this direction from [3], the set ${\mathcal{S}}$ of “free” positions has to be very massive, namely its cardinality has to satisfy $T\geq(1-\kappa)S$ for some small (and unspecified) absolute constant $\kappa>0$. In the case of square-free numbers instead of prime numbers, a similar result has been obtained in [6] with any fixed $\kappa<2/5$ (one can also find in [6] some results on the distribution of the value of the Euler function and quadratic non-residues in ${\mathcal{N}}(a;{\mathcal{S}})$). Here we address some problems at the other extreme, and relax the strength of arithmetic conditions on the elements from ${\mathcal{N}}(a;{\mathcal{S}})$ but instead consider much sparse sets ${\mathcal{S}}$ of available positions.

2. Main results

Throughout the paper, the letter $p$ always denotes a prime number.

As usual $A=O(B)$, $A\ll B$, $B\gg A$ are all equivalent to $|A|\leq c|B|$ for some absolute constant $c>0$, whereas $A=o(B)$ means that $A/B\to 0$.

We say that a pair $(\alpha,\beta)$ is admissible if for any prime $p$ and any integer $\lambda$ with $\gcd(\lambda,p)=1$ we have

[TABLE]

as $p\to\infty$, where $t$ is the multiplicative order of $\lambda$ modulo $p$.

Concerning admissible pairs, Korobov [16] has shown that the pair

[TABLE]

is admissible. For shorter ranges of $t,$ Korobov’s bound has been improved by Heath-Brown and Konyagin [12] who show that the pairs

[TABLE]

and

[TABLE]

are admissible.

More recently Shkredov [19, 20] has shown that the pair

[TABLE]

is admissible, which improves on the pairs (2.1) and (2.2) in the medium range of $t$.

Furthermore, the truly remarkable result of Bourgain, Glibichuk and Konyagin [5] implies that for any $\zeta>0$ there is some $\vartheta>0$ that depends only on $\zeta$ such that

[TABLE]

is admissible.

Our first result is as follows.

Theorem 2.1**.**

Suppose that for an admissible pair $(\alpha,\beta)$ and some positive numbers $\eta$ and $\delta$, we have

[TABLE]

Suppose further that $S$, $T$ and $X$ are parameters satisfying

[TABLE]

Let $\Delta>1$ and integer $k\geq 1$ satisfy

[TABLE]

Then for any sequence of complex numbers $\Gamma=\{\gamma_{n}\}_{n=1}^{T}$ with $|\gamma_{n}|\leq 1$ and integers ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ with $0\leq s_{1}<\ldots<s_{T}\leq S$ we have

[TABLE]

We note that under (2.6) the condition (2.7) also follows from a simpler inequality

[TABLE]

Considering the strength of Theorem 2.1, we take $\Delta=X^{1/2}$ and $T=X^{1+\varepsilon}$. Using the admissible pair of Heath-Brown and Konyagin (2.1), we obtain a power saving in Theorem 2.1 provided $S\leq X^{7/6-\varepsilon}$, improving of Garaev’s range of $S\leq X^{15/14-\varepsilon}$. With the same choice of parameters and using the admissible pair of Bourgain, Glibichuk and Konyagin (2.4), we obtain a power saving in Theorem 2.1 provided $S\leq X^{3/2-\varepsilon}$.

Using a different method we can set $\Delta=1$ and also extend the range of $S$ for which we may obtain a nontrivial bound for $V_{\lambda}(\Gamma,{\mathcal{S}};T,X)$ at the cost of making the power saving explicit.

Theorem 2.2**.**

There exists some absolute constant $\rho>0$ such that

[TABLE]

Comparing the bound of Theorem 2.2 with the trivial bound $XT^{2}$, we see that it is nontrivial provided

[TABLE]

which on taking $T=X^{1+\varepsilon}$, we obtain a power saving in Theorem 2.2 provided $S\leq T^{2-\varepsilon}$.

For a sequence of points ${\mathcal{A}}=\{a_{n}\}_{n=1}^{T}$ we define the discrepancy $D$ of ${\mathcal{A}}$ by

[TABLE]

where $A(a,b)$ denotes the number of points of ${\mathcal{A}}$ falling in the interval $[a,b]\in[0,1]$. Garaev [9] combines his bound for $V_{\lambda}(\Gamma,{\mathcal{S}},T,X,\Delta)$ with a result of Erdös and Murty [8], ††margin:

I: Expanded with (2.8), split one sentence into 2

which in particular implies that

[TABLE]

and the Erdös-Turán inequality (see for example [7]). This allows Garaev [9, Section 3] to show that for any $\varepsilon>0$ there is some $\delta>0$ such that for almost all primes $p\leq X$, the sequence

[TABLE]

with $T=\left\lceil X(\log{X})^{2+\varepsilon}\right\rceil$, has discrepancy

[TABLE]

provided $S\leq X^{15/14+o(1)}$ as $X\to\infty$.

For comparison with our bound, Theorem 2.2 produces the following result. For any $\varepsilon>0$ and almost all primes $p\leq X$, the sequence (2.9) with $T=\left\lceil X^{1+\varepsilon}\right\rceil$ has discrepancy

[TABLE]

provided $S\leq X^{2-\varepsilon}$ as $X\rightarrow\infty$.

We now give an application of Theorem 2.2 to the numbers with prescribed digits, namely to the integers from the set ${\mathcal{N}}(a;{\mathcal{S}})$, defined in Section 1. We denote by $\omega(k)$ the number of distinct prime divisors of an integer $k\geq 1$.

Theorem 2.3**.**

Let us fix some $\varepsilon>0$. For any sequence of integers ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ with $0\leq s_{1}<\ldots<s_{T}\leq S$ with

[TABLE]

and any $S$-bit integer $a$, we have

[TABLE]

for some $\delta>0$ which depends only on $\varepsilon$.

3. Preliminary results

We recall that $A\ll B$ and $A=O(B)$ are both equivalent to the inequality $|A|\leq cB$ for some constant $c$, which throughout the paper may depend on $q$ and occasionally, where obvious, on the integer parameter $k\geq 1$.

We alslo use $\Sigma^{*}$ to indicate that the summation is taken over a reduced residue system. That is, for any function $\psi$ and integer $k$, we have

[TABLE]

We need the following simplified form of the large sieve inequality, see [13, Theorem 7.11].

Lemma 3.1**.**

For any $K\geq 1$ and increasing sequence of integers ${\mathcal{S}}=\{s_{n}\}_{n=1}^{T}$ with $\max_{s\in{\mathcal{S}}}s=S$, we have

[TABLE]

The following is [15, Theorem 5.5].

Lemma 3.2**.**

For each integer $t$ and prime $\ell\equiv 1\bmod t$ we fix some element $g_{t,\ell}$ of multiplicative order $t$ modulo $\ell$. Then, for any fixed integer $k\geq 2$ and an arbitrary $U>1$, the bound

[TABLE]

holds for all primes $\ell\equiv 1\bmod t$ except at most $U/\log{U}$ of them.

Lemma 3.3**.**

Let $\lambda$ be a fixed integer. For any $Z>0$ we have

[TABLE]

Proof.

If $\mathrm{ord}_{p}\,{\lambda}=y$ then $\lambda^{y}-1\equiv 0\bmod{p}$. This implies that

[TABLE]

where as before, $\omega(k)$ denotes the number of distinct prime divisors of an integer $k\geq 1$. Hence,

[TABLE]

which gives the desired result.     \sqcap$$\sqcup

The following is a special case of [4, Corollary 4.2].

Lemma 3.4**.**

Let $p_{1}$ and $p_{2}$ be primes and let ${\mathcal{H}}$ be a subgroup of ${\mathbb{Z}}_{q}^{*}$, where $q=p_{1}p_{2}$ such that

[TABLE]

for some fixed $\delta>0$. Then

[TABLE]

for some $\varrho>0$ which depends only on $\delta>0$.

4. Proof of Theorem 2.1

4.1. Initial tranformations

Let

[TABLE]

It is also convenient to define $a_{p}$ as any integer $a\in\{1,\ldots,p-1\}$ with

[TABLE]

so that

[TABLE]

However, it is more convenient to work with the sums where each term is divided by the divisor function $\tau(p-1)$. We define

[TABLE]

and note the inequality $\tau(n)=n^{o(1)}$ implies that

[TABLE]

Hence it is enough to prove

[TABLE]

where $\alpha,\beta,\delta,\eta$ satisfy (2.5) and $(\alpha,\beta)$ is an admissible pair.

Fix some $p\leq X$ and consider $\sigma_{p}(a_{p})$. We split $s_{n}$ into arithmetic progressions mod $t_{p}$. Using the orthogonality of exponential functions, we obtain

[TABLE]

and hence

[TABLE]

Let $\xi>0$ be a real parameter to be chosen later. We set

[TABLE]

and partition summation over $d$ according to $D_{p}$. This gives

[TABLE]

where

[TABLE]

and

[TABLE]

The equation (4.3) implies that

[TABLE]

which on averaging over $p\leq X$ gives

[TABLE]

where

[TABLE]

4.2. The sum $\Sigma_{1}$

To bound $\Sigma_{1}$ we use the argument of Garaev [9, Theorem 3.1]. Fix some $p\leq X$ and consider $\sigma_{p,1}(a_{p})$. From (4.4) and the Cauchy-Schwarz inequality

[TABLE]

Expanding the square and interchanging summation gives

[TABLE]

By the orthogonality of exponential functions, the inner sum vanishes unless $b_{1}=b_{2}$. Hence

[TABLE]

where we have used the inequality

[TABLE]

since $t_{p}\mid(p-1)$. Summing over $p\leq X$ we see that

[TABLE]

We define the sequence of numbers $X_{j}$ for $1\leq j\leq J$, where

[TABLE]

by

[TABLE]

and partition the set of primes $p\leq X$ into the sets

[TABLE]

Writing

[TABLE]

we have

[TABLE]

For each integer $r$, we define the set ${\mathcal{Q}}(r)$ by

[TABLE]

so that, replacing $t_{p}$ with $r$ for $p\in{\mathcal{Q}}(r)$, we obtain

[TABLE]

For each prime $p\in{\mathcal{Q}}(r)$ we have $r\mid(p-1)$ and hence for $X_{j}\leq r<2X_{j}$ we also have

[TABLE]

This implies that

[TABLE]

and hence

[TABLE]

where $F_{j}(d)$ is given by

[TABLE]

An application of Lemma 3.1 gives

[TABLE]

which combined with (4.12) implies that

[TABLE]

and hence by (4.10)

[TABLE]

4.3. The sum $\Sigma_{2}$

Fix some $p\leq X$ and consider $\sigma_{p,2}(a_{p})$. For each value of $d$ in the outermost summation we split summation over $x$ into arithmetic progressions mod $t_{p}/d$. Recalling that $\sigma_{p,2}(a_{p})$ is given by

[TABLE]

we see that

[TABLE]

and hence

[TABLE]

where $f_{d,p}$ is chosen to satisfy

[TABLE]

Let

[TABLE]

so that

[TABLE]

We consider bounding the terms $U(p,d)$. By the Cauchy-Schwarz inequality

[TABLE]

Using the orthogonality of exponential functions again, we see that the last sums vanishes unless $b_{1}=b_{2}$. This gives

[TABLE]

After rearranging and extending the summation over $b$ to the complete residue system modulo $t_{p}/d$, we derive

[TABLE]

where for an integer $r\geq 1$ we define

[TABLE]

Substituting this in (4.14) gives

[TABLE]

Summing over $p\leq X$ gives

[TABLE]

which by the Cauchy-Schwarz inequality implies that

[TABLE]

At this point our strategy is to rearrange summation so we may apply Lemma 3.2. We define the sequence $X_{j}$ as in (4.8), we let ${\mathcal{Q}}(r)$ be given by (4.11) and for each integer $r$ we define the following subsets $S_{i}(r)$ of ${\mathcal{Q}}(r)$

[TABLE]

Writing

[TABLE]

the above implies that

[TABLE]

To further transform the sums $\Sigma_{2,i,j},$ define the numbers $Z_{j}$ by

[TABLE]

so that

[TABLE]

After interchanging summation, we arrive at

[TABLE]

Let $\rho$ be a parameter to be chosen later. We now partition summation over $i$ and $j$ in $\Sigma_{2}$ as follows

[TABLE]

where

[TABLE]

To estimate $\Sigma_{2}^{\leq}$, we first fix some $j$ with $X_{j}\leq 2^{i\rho}$. Considering the inner summation over $p$, we partition ${\mathcal{S}}_{i}(r)$ according to Lemma 3.2. Let

[TABLE]

and for integer $k$ we define the sets $S^{(1)}_{i}(r)$ and $S^{(2)}_{i}(r)$ by

[TABLE]

Lemma 3.2 implies that

[TABLE]

Considering ${\mathcal{S}}^{(1)}_{i}(r)$ and using the fact that $r\mid p-1$ for $p\in{\mathcal{S}}_{i}(r)$ gives

[TABLE]

which implies that

[TABLE]

Recalling the choice of $U_{i}(r)$ we see that

[TABLE]

which on assuming that

[TABLE]

simplifies to

[TABLE]

Hence considering $\Sigma_{2,i,j}$, we have

[TABLE]

after the change of variable $r\to dr$. Writing

[TABLE]

the above implies

[TABLE]

Considering the sum $W_{j}(d)$ and recalling the definition of $V(r)$ given by (4.15), we have

[TABLE]

Considering the last sum on the right, we have

[TABLE]

Since the term

[TABLE]

is bounded by the number of divisors of $s_{n_{2}}-s_{n_{1}}$, we see that

[TABLE]

and hence

[TABLE]

which gives

[TABLE]

Substituting the above into (4.22) we get

[TABLE]

which simplifies to

[TABLE]

on recalling the choice of $Z_{j}$ given by (4.16).

We now assume that

[TABLE]

Without loss of generality, we can also assume that $S=X^{O(1)}$ and thus $(SX)^{o(1)}=X^{o(1)}$. Hence, the above bounds further simplify to

[TABLE]

Summing over $i$ and $j$ with $X_{j}\leq 2^{i\rho}$ we arrive at

[TABLE]

and hence

[TABLE]

We next consider $\Sigma_{2}^{\geq}$. We begin our treatment of $\Sigma_{2}^{\geq}$ in a similar fashion to $\Sigma_{2}^{\leq}$. In particular, we use (4.17) and the assumption that $(\alpha,\beta)$ is admissible to obtain

[TABLE]

as $i\to\infty$.

Using (4.19) and then rearranging the order of summation, the above reduces to

[TABLE]

where

[TABLE]

We see from the definition (4.15) that

[TABLE]

and hence

[TABLE]

Since obviously $S\leq X^{O(1)}$, we can replace both $2^{o(i)}$ and $S^{o(1)}$ with $X^{o(1)}$. Recalling the choice of $Z_{j}$ and the assumption (4.24), we get

[TABLE]

This implies that

[TABLE]

Substituting the bounds (4.25) and (4.27) in (4.18), we see that

[TABLE]

4.4. Concluding the proof

Substituting (4.13) and (4.28) in (4.5), gives

[TABLE]

Let $\eta>0$ be a parameter and make the substitution

[TABLE]

The above transforms into

[TABLE]

Next we chooise

[TABLE]

to balance the second and fourth terms. This gives

[TABLE]

We now note that the assumption (2.5) implies that

[TABLE]

which is the desired bound.

Finally, to complete the proof, it remains to note that (4.20) is satisfied by the assumption (2.7) and (4.24) is satisfied by (2.6).

5. Proof of Theorem 2.2

5.1. Initial tranformations

As before, for each prime $p$ we define the number $a_{p}$ by (4.1). Taking $Z=X^{1/4}$ in Lemma 3.3 and recalling that $t_{p}$ denotes the order of $\lambda$ mod $p$, we have

[TABLE]

We define the sequence of numbers $X_{j}$, as in (4.8) with $\Delta=X^{1/4}$. We also define the sets ${\mathcal{R}}_{j}$ as in (4.9) for $j=1,\ldots,J$ with $J$ given by (4.7).

Hence, partitioning summation over $p$ in (5.1) according to ${\mathcal{R}}_{j}$ gives,

[TABLE]

where

[TABLE]

We define the number $Y$ by

[TABLE]

and let $I$ be the largest integer $j$ with $X_{j}\leq Y$ (since $S\geq T$ we obviously have $Y\geq X^{3/4}>X^{1/4}$ so $I$ is correctly defined).

We now further partition the summation over $j$ and re-write (5.1) as

[TABLE]

where

[TABLE]

5.2. The sum $W^{\leq}$

We fix some $j$ with $X^{1/4}\leq X_{j}<Y$. Considering $W_{j}$, we define the sets

[TABLE]

so that

[TABLE]

where $U_{j,r}$ is given by

[TABLE]

For each $p\in{\mathcal{V}}_{j}(r)$ we define the complex number $c_{j,r,p}$ by

[TABLE]

so that

[TABLE]

and writing

[TABLE]

we see that

[TABLE]

We have

[TABLE]

where

[TABLE]

and hence by the Cauchy-Schwarz inequality

[TABLE]

Expanding the square and interchanging summation gives

[TABLE]

which implies that

[TABLE]

Since

[TABLE]

the set

[TABLE]

is a subgroup of ${\mathbb{Z}}^{*}_{p_{1}p_{2}}$ and from the inequalities

[TABLE]

we see that the conditions of Lemma 3.4 are satisfied. An application of Lemma 3.4 gives

[TABLE]

which by the Cauchy-Schwarz inequality implies that

[TABLE]

and hence by (5.7)

[TABLE]

Since

[TABLE]

we get

[TABLE]

Recalling (5.9) and the assumption each $|\gamma_{n}|\leq 1$, we see that

[TABLE]

where $V(r)$ is defined by (4.15). By (5.11) we have

[TABLE]

and hence by (5.8)

[TABLE]

Combining the above with (5.6) gives

[TABLE]

As in the proof of Theorem 2.1, see (4.23), we have

[TABLE]

where we have used the assumption $S\leq T^{2}$ and $T>X$ as otherwise Theorem 2.2 is trivial. Substituting the above into (5.12) gives

[TABLE]

and hence by (5.4)

[TABLE]

5.3. The sum $W^{\geq}$

We fix some $j$ with $Y\leq X_{j}\leq X$ and arrange $W_{j}$ as follows

[TABLE]

and hence there exists some sequence of complex numbers $c_{j,p}$ with $|c_{j,p}|=1$ such that

[TABLE]

An application of the Cauchy-Schwarz inequality gives

[TABLE]

Since the sequence $s_{n}$ is increasing and bounded by $S$, we see that

[TABLE]

so that writing

[TABLE]

the above implies

[TABLE]

Considering ${\mathbf{W}}_{j}$, expanding the square and interchanging summation gives

[TABLE]

for some integers $a_{p_{1}p_{2}}$ with $\gcd(a_{p_{1}p_{2}},p_{1}p_{2})=1$. By (5.5) and (5.10)

[TABLE]

and hence

[TABLE]

where

[TABLE]

Considering $Z(p_{1},p_{2})$, by the Cauchy-Schwarz inequality, we have

[TABLE]

Now, since

[TABLE]

we see that

[TABLE]

Since $t_{p_{1}},t_{p_{2}}\geq X_{j}$, we have

[TABLE]

which implies

[TABLE]

which after substituting the above in (5.15) gives

[TABLE]

We have

[TABLE]

and

[TABLE]

so that

[TABLE]

Combining the above with (5.14) gives

[TABLE]

which simplifies to

[TABLE]

since we may assume $S\leq X^{2+o(1)}$. By (5.4) we have

[TABLE]

5.4. Concluding the proof

Substituing (5.13) and (5.16) in (5.3) we derive

[TABLE]

Recalling the choice of $Y$ in (5.2) the above simplifies to

[TABLE]

and the result follows with $\rho=\varrho/4$ (as clearly $\varrho\leq 1$ and thus $\rho<1/2$)

6. Proof of Theorem 2.3

First we note that without loss of generality we may assume the binary digits of $a$ are zeros on all positions $j\in{\mathcal{S}}$.

For a prime $p$, let $N_{p}(a;{\mathcal{S}})$ be the number of $z\in{\mathcal{N}}(a;{\mathcal{S}})$ with $p\mid z$. One can easily see that $N_{p}(a;{\mathcal{S}})$ is the number of solutions to the congruence

[TABLE]

We now proceed similarly to the proof of [15, Theorem 18.1]. Using the orthogonality of exponential functions, we write

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

Using [15, Equation (18.2)] we write

[TABLE]

where

[TABLE]

Now, by Theorem 2.2 if we fix some $\varepsilon_{0}>0$, then there is some $\kappa>0$ such that if

[TABLE]

then we have

[TABLE]

Since $S\leq T^{2-\varepsilon}$, to satisfy the above conditions, it is enough to define $\varepsilon_{0}$ by the equation

[TABLE]

or, more explicitely,

[TABLE]

Combining this with (2.8), we see that for all but $o(X/\log X)$ primes $p\leq X$ we have $M_{p}\leq TX^{-\kappa/3}$. For each of these primes $p$, a combination of (6.1) and (6.2) implies that $N_{p}(a;{\mathcal{S}})>0$ (provided that $p$ is large enough), which concludes the proof.

7. Possible improvements

We note that one can get an improvement of Theorem 2.1 by using a combination of different admissible pairs depending on the range of $d$ in our treatement of the sum (4.17) in and thus making the choice of $\alpha$ and $\beta$ in (4.26) dependent on $i$ and $j$.

In particular, one can use the admissible pairs (2.1), (2.2), (2.3) and (2.4) as well the admissible pairs of Konyagin [14] and Shteinikov [23] for small values of $d$ in (4.17).

Acknowledgement

This work was partially supported by the NSF Grant DMS 1600154 (for M.-C. C.) and by ARC Grant DP170100786 (for I. S.).