The discrepancy of $(n_kx)$ with respect to certain probability measures

TL;DR

This paper proves a conjecture regarding the discrepancy of lacunary sequences with respect to certain probability measures, showing almost sure bounds on the normalized discrepancy and improving previous results on related Diophantine approximation products.

Contribution

It establishes almost sure bounds on the discrepancy of lacunary sequences under specific measures, confirming a conjecture and refining earlier results in Diophantine approximation.

Findings

Almost sure bounds on the discrepancy for lacunary sequences

Validation of a conjecture by Haynes, Jensen, and Kristensen

Improved bounds on products involving Diophantine approximation

Abstract

Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\|q\alpha\| \|q\beta-\gamma\| $.

Equations

$$\frac{1}{4}\le \underset{N\to \infty }{\mathrm{limsup}}\frac{N{D}_N(n_{k}x)}{\sqrt{N\log \log N}}\le C$$

$$D_N(x_n)=\underset{0\le \alpha <\beta <1}{\sup }\left|\frac{1}{N}Z(N;\alpha ,\beta )-(\beta -\alpha )\right|$$

$$D_N(x_n)\ge c\frac{\log N}{N}\text{for inf. many }N=1,2,\dots$$

$$\frac{n_{k+1}}{n_k}\ge q,k=1,2,\dots$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{\left|\sum _{k=1}^{N}e(n_{k}x)\right|}{\sqrt{N\log \log N}}=1\text{ for Lebesgue-almost all }x\in [0,1),$$

$$\frac{1}{4}\le \underset{N\to \infty }{\mathrm{limsup}}\frac{N{D}_N(n_{k}x)}{\sqrt{N\log \log N}}\le C_q,$$

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{N{D}_N(X_1,X_2,\dots ,X_N)}{\sqrt{2N\log \log N}}=\frac{1}{2}$$

$$\hat{\mu }(t)=\int e^{2\pi ixt}\mathrm{d}\mu (x),t\in \mathbb{R}$$

$$|\hat{\mu }(t)|\ll |t{|}^{-\eta },|t|\to \infty$$

$$\sum _{N=1}^{\infty }\frac{1}{N^3}\sum _{m,n=1}^N\hat{\mu }(h(q_m-q_n))<\infty \text{vspace}-1mm$$

$$\frac{1}{4}\le \underset{N\to \infty }{\mathrm{limsup}}\frac{N{D}_N(n_{k}x)}{\sqrt{N\log \log N}}\le C\text{ for μ-almost all }x\in [0,1),$$

$$q\Vert q{\alpha }_i\Vert \Vert q\beta -\gamma \Vert <1/(\log q{)}^{\frac{1}{2}-\epsilon }\text{ for inf. many }q=1,2,\dots$$

$$q\Vert q{\alpha }_j\Vert \Vert q\beta -\gamma \Vert <\frac{({\log }_{3}q{)}^{1/2+\epsilon }}{(\log q{)}^{1/2}}\text{ for inf. many }q=1,2,\dots$$

$${\dim }_{F}F:=\sup \left\{0\le \eta \le 1:\exists \mu \in M_1^{+}(F)\text{ with }\hat{\mu }(t)=O(|t{|}^{-\eta /2}),|t|\to \infty \right\}.\text{vspace}2mm$$

$$A=\left\{x\in [0,1):\underset{N\to \infty }{\mathrm{limsup}}{N}^{1/2}(\log \log N{)}^{-1/2}{D}_N(n_{k}x)=0\text{ or }\infty \right\}$$

$$N{D}_N(n_{k}x)=\underset{0\le \alpha <\beta <1}{\sup }\left|\sum _{k\le N}{\chi }_{[\alpha ,\beta )}(n_{k}x)-N(\beta -\alpha )\right|\le 2\underset{f\in \mathcal{F}}{\sup }\left|\sum _{k\le N}f(n_{k}x)\right|,$$

$$\text{Var}f\le 2,\Vert f{\Vert }_{\infty }\le 1,f(x+1)=f(x)\text{ and }{\int }_0^{1}f(x)\mathrm{d}x=0$$

$$f(x)=\sum _{j=1}^{\infty }2c_j\cos (2\pi jx)=\sum _{|j|=1}^{\infty }{c}_{j}e(jx)$$

$$f_n(x)=\sum _{1\le |j|\le n}{c}_{j}e(jx).$$

$${\varphi }_n(x)=f(x)-f_n(x),$$

$$\psi (x;m)=\sum _{r^m\le |j|<r^{m+1}}{c}_{j}e(jx),$$

$${\Phi }_N(x;m)=\sum _{k\le N}\psi (n_{k}x;m).$$

$$\int |{\Phi }_N(x;m){|}^2\mathrm{d}\mu (x)\ll \frac{N}{r^{m\eta }},$$

$$\int |{\Phi }_N(x;m){|}^2\mathrm{d}\mu (x)$$

$$\ll N\log \left(\frac{r^{m+1}}{r^m}\right)\frac{1}{r^m}\ll \frac{N}{r^m}.$$

$${{{\sum \sum }_{1\le k,l\le N}}_{r^m\le |i|,|j|<r^{m+1}}}_{|i{n}_k-j{n}_l|\ge 1}\frac{c_i{c}_j}{|i{n}_k-j{n}_l{|}^{\eta }}\ll {{{\sum \sum }_{1\le k\le N}}_{r^m\le |i|,|j|<r^{m+1}}}_{i\ne j}{c}_i{c}_j|n_k(i-j){|}^{-\eta }+{{{\sum \sum }_{1\le l<k\le N}}_{r^m\le |i|,|j|<r^{m+1}}}_{|i{n}_k-j{n}_l|\ge 1}\frac{c_i{c}_j}{|i{n}_k-j{n}_l{|}^{\eta }}\cdot$$

$${{{\sum \sum }_{1\le k\le N}}_{r^m\le i,j<r^{m+1}}}_{i\ne j}{c}_i{c}_j{\left(n_k\left|i-j\right|\right)}^{-\eta }$$

$$|i{n}_k-j{n}_l|=|i|n_k\left|1-\frac{j{n}_l}{i{n}_k}\right|\ge |i|n_k\left|1-\frac{1}{r}\right|\gg r^m{n}_k,$$

$${{\sum \sum }_{1\le l<k\le N}}_{r^m\le |i|,|j|<r^{m+1}}{c}_i{c}_j|i{n}_k-j{n}_l{|}^{-\eta }$$

$$\int {\left(\sum _{k\le N}{\varphi }_T(n_{k}x)\right)}^2\mathrm{d}\mu (x)\ll \frac{N}{T^{\eta }}\cdot$$

$${\varphi }_T(n_{k}x)$$

Full text

[TABLE]

for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood Conjecture.

1 Introduction

Let $(x_{n})_{n=1}^{\infty}$ be a sequence of numbers in the unit interval $[0,1)$. We define the $N$-discrepancy of the sequence $(x_{n})_{n=1}^{\infty}$ to be

[TABLE]

where $Z(N;\alpha,\beta):=\#\{1\!\leq k\!\leq\!N\,:\,\alpha\leq x_{k}\leq\beta\}$. A sequence $(x_{n})_{n=1}^{\infty}$ is by definition uniformly distributed $\mod 1$ if and only if $D_{N}(x_{n})\to 0$ as $N\to\infty$. Regarding the order of magnitude of the discrepancy of arbitrary sequences, Schmidt [24] has shown that the discrepancy of any sequence $(x_{n})_{n=1}^{\infty}\subseteq[0,1)$ satisfies

[TABLE]

where $c>0$ is an absolute constant; thus the discrepancy of an arbitrary sequence cannot tend to [math] arbitrarily fast.

A case of particular interest is the discrepancy of $(n_{k}x)_{k=1}^{\infty}$, where $(n_{k})_{k=1}^{\infty}$ is lacunary and $x\in[0,1)$. Recall that a sequence $(n_{k})_{k=1}^{\infty}$ of positive integers is called lacunary if there exists some constant $q>1$ such that

[TABLE]

It is well known that whenever (1) holds, the sequence of functions $(e(n_{k}x))_{k=1}^{\infty}$ behaves like a sequence of independent random variables (here and in what follows we use the notation $e(x)=e^{2\pi ix}$); for more details we refer to the survey papers [3, 15]. One instance of this phenomenon is a result of Erdős and Gal [6] stating that

[TABLE]

which is an analogue of the Law of the Iterated Logarithm for sequences of independent random variables. Regarding the precise order of magnitude of $D_{N}(n_{k}x)$ in that case, it had been conjectured that $D_{N}(n_{k}x)$ also satisfies a Law of the Iterated Logarithm, and this was shown to be true by Philipp [20].

**Theorem (Philipp)​​​ **.

Let $(n_{k})_{k=1}^{\infty}$ be a lacunary sequence of integers such that (1) is satisfied. Then for Lebesgue-almost all $x\in[0,1)$ we have

[TABLE]

where $C_{q}\leq 166+664(q^{1/2}-1)^{-1}$ is a constant which depends on $q>1$.

This is in accordance with the Chung-Smirnov Law of the Iterated Logarithm, which states that for any sequence $(X_{n})_{n=1}^{\infty}$ of independent random variables, uniformly distributed on $[0,1)$, we have

[TABLE]

with probability $1$ (see [25, p.504]), thus further indicating the resemblance of $(n_{k}x)_{k=1}^{\infty}$ with a sequence of independent random variables. The exact value of the limsup in (3) for specific choices of the sequence $(n_{k})_{k=1}^{\infty}$ has been calculated by Fukuyama et.al. in a series of papers [8, 9, 10, 11, 12].

In the present article we examine whether Philipp’s metrical result can be generalised for measures which are supported on several fractal subsets of the unit interval. We focus our attention on probability measures $\mu$ such that their Fourier transform defined by

[TABLE]

has a prescribed decay rate. In the results to follow, we assume that the Fourier transform of $\mu$ has a polynomial decay rate, that is, an asymptotic relation of the form

[TABLE]

holds for some constant $\eta>0$. The connection of the decay rate of the Fourier transform of $\mu$ with distribution properties is not unexpected, in view of the following theorem of Davenport, Erdős and LeVeque.

**Theorem (Davenport, Erdős & LeVeque)​​ **.

Let $\mu$ be a probability measure supported on $[0,1]$ and $(q_{n})_{n=1}^{\infty}$ be a sequence of natural numbers. If

[TABLE]

for all integers $h\neq 0$, then the sequence $(q_{n}x)_{n\in\mathbb{N}}$ is uniformly distributed modulo one for $\mu$–almost all $x\in[0,1)$.

The main result of this paper is the following.

Theorem 1**.**

Let $(n_{k})_{k=1}^{\infty}$ be a lacunary sequence of integers satisfying (1). Assume $\mu$ is a probability measure on $[0,1)$ such that (4) holds for some $\eta>0.$ Then the discrepancy $D_{N}(n_{k}x)$ satisfies

[TABLE]

where the constant $C>0$ only depends on the value of $q>1$ as in (1). Additionally $C\leq 166+664(q^{1/2}-1)^{-1}.$

An application of Theorem 1 is an improvement of a result of Haynes, Jensen and Kristensen in [14] relevant to an inhomogeneous version of Littlewood’s conjecture, which is the statement that for all $\alpha,\beta\in\mathbb{R}$, we have $\liminf q\|q\alpha\|\|q\beta\|=0.$ This is clearly the case when $\alpha$ or $\beta$ is an element of the set ${\rm{\bf Bad}}:=\{x\in[0,1):\liminf q\|qx\|>0\}$ of badly approximable numbers. The result proved in [14] is the following:

**Theorem (Haynes, Jensen & Kristensen)​​ **.

Fix $\varepsilon>0$ and a sequence $(\alpha_{i})_{i=1}^{\infty}\subseteq{\rm{\bf Bad}}$. Then there exists a set $G\subseteq{\rm{\bf Bad}}$ of Hausdorff dimension $\dim G=1$, such that for all $\beta\in G$ the following holds:

[TABLE]

for all $i\geq 1$ and all $\gamma\in\mathbb{R}$.

The proof of the result in [14] relies on a metric discrepancy estimate with respect to certain probability measures supported on subsets of the set ${\rm{\bf Bad}}$. More precisely, if $F_{N}:=\{x=[a_{1},a_{2},\ldots]\in[0,1):a_{n}\leq N\text{ for all }n\geq 1\}$ is the set of $x\in[0,1)$ such that the partial quotients in the continued fraction expansion of $x$ are at most equal to $N$, a theorem of Kaufman [17], later improved by Queffélec and Ramaré [23], states that the sets $F_{N},\,N\geq 2$ support probability measures with two key properties:

**Theorem (Kaufman & Queffélec-Ramaré)​​ **.

Let $N\geq 2$. If $\varepsilon>0$ and $\frac{1}{2}<\delta<\dim F_{N}$, then the set $F_{N}$ supports a probability measure $\mu=\mu(N,\delta,\varepsilon)$ with the following properties:

(i) $\mu(I)\leq c_{1}|I|^{\delta}$ for any interval $I\subseteq[0,1)$, and

(ii) $|\widehat{\mu}(t)|\leq c_{2}(1+|t|)^{-\eta+8\varepsilon}$  for all $t\in\mathbb{R},$ where $\eta=\frac{\delta(2\delta-1)}{(2\delta+1)(4-\delta)}>0$.

*Here $c_{1},c_{2}>0$ are absolute constants. *

In the current paper, adapting the method of proof in [14] together with the sharper discrepancy estimate coming from Theorem 1, we are able to obtain a slight improvement to the result of Haynes, Jensen and Kristensen:

Theorem 2**.**

Let $\varepsilon>0$ be fixed and $(\alpha_{j})_{j=1}^{\infty}\subseteq{\rm{\bf Bad}}$ be a sequence of badly approximable numbers. There exists a subset $G\subseteq{\rm{\bf Bad}}$ of Hausdorff dimension $\dim G=\dim{\rm{\bf Bad}}=1$ such that for any $\beta\in G,$ the following holds:

[TABLE]

for all $j=1,2,\ldots$ and for all $\gamma\in[0,1)$.

As another consequence of Theorem 1, we obtain a statement regarding the Fourier dimension of the set of exceptions to Philipp’s theorem, namely the set of $x\in[0,1)$ for which (3) fails. Given a subset $F\subseteq[0,1)$, the Fourier dimension of $F$ is defined by

[TABLE]

where $M_{1}^{+}(F)$ denotes the set of all positive Borel probability measures with support in $F$. A result of Frostman states that whenever a set $F\subseteq[0,1)$ supports a probability measure $\mu$ such that $\widehat{\mu}(t)=O(|t|^{-\eta/2})$, then its Hausdorff dimension is $\dim F\geq\min\{1,\eta\}$ (see [7, Chapter 4]). Therefore, the Fourier dimension of an arbitrary set is bounded above by the Hausdorff dimension: $\dim_{F}F\leq\dim F$.

Corollary 3**.**

Let $(n_{k})_{k=1}^{\infty}$ be a lacunary sequence of integers and consider the set

[TABLE]

of $x\in[0,1)$ for which the conclusion (3) in Philipp’s Theorem fails. Then the Fourier dimension of $A$ is $\dim_{F}A=0$.

The proof of Corollary 3 is straightforward: suppose $\dim_{F}A>0$. Then $A$ supports some probability measure $\mu$ such that $\widehat{\mu}(t)=O(|t|^{-\eta/2}),\,|t|\to\infty$ for some $\eta>0$, and by Theorem 1, (6) holds for $\mu$–almost all $x\in A$, which contradicts the definition of $A$.

However, regarding the Hausdorff dimension of the set $A$ defined in Corollary 3, it can be deduced that $\dim A=1$. Indeed, given a lacunary sequence $(n_{k})_{k=1}^{\infty}$, a result obtained independently by Pollington [21] and de Mathan [19] states that the set $B:=\{x\in[0,1):(n_{k}x)_{k=1}^{\infty}\text{ is not dense}\mod 1\}$ has Hausdorff dimension $1$. Thus $\dim A=1$ as well, since $B\subseteq A$.

2 Proof of Theorem 1

2.1 The upper bound

We employ a classical method of proof for discrepancy estimates, which has been used by Philipp [20], Erdős $\&$ Gal [6] and Gal $\&$ Gal [13]. For any positive integer $N$, the discrepancy $D_{N}(n_{k}x)$ satisfies

[TABLE]

where $\mathcal{F}$ denotes the set of functions $f:[0,1)\rightarrow\mathbb{R}$ which are $1$-periodic with $\int_{0}^{1}\!f(x)\mathrm{d}x=0$ and have bounded variation. Moreover, since every such function is trivially the sum of an even and an odd function with the same properties, we may restrict our attention to the set $\mathcal{F}^{*}\subseteq\mathcal{F}$ of functions $f\in\mathcal{F}$ which are additionally even.

2.1.1 Some auxiliary results

Let $f$ be an even function of bounded variation on $[0,1]$ such that

[TABLE]

and let

[TABLE]

be its Fourier series expansion. Observe that $c_{j}=c_{-j}$ for all $j$ and (8) imposes $c_{0}=0$ and $c_{j}\leq|j|^{-1}$. We set

[TABLE]

Write $r=q^{1/2}$ (where $q>1$ is as in (1)) and define

[TABLE]

Also for $f\in L_{2}(\mu)$ we put $\|f\|_{L_{2}(\mu)}^{2}=\int|f|^{2}\mathrm{d}\mu.$ In the following lemmas we calculate the $\|\cdot\|_{L_{2}(\mu)}$–norm of sums of the form $\sum_{k\leq N}\phi_{T}(n_{k}x)$, first by calculating the norm of the blocks $\Phi_{N}(x;m)$ in Lemma 4, and then combining these estimates in Lemma 5.

Lemma 4**.**

We have

[TABLE]

where $\eta>0$ is as in (4).

Proof.

We calculate

[TABLE]

Regarding the first of these terms, we can show as in [20, Lemma $1$] that it has order of magnitude

[TABLE]

The second term is

[TABLE]

The first of the sums in the right hand side of (9) is at most

[TABLE]

Regarding the second sum in the right hand side of (9), under the conditions of summation we get

[TABLE]

whence

[TABLE]

∎

Lemma 5**.**

For any positive integer $N,T$ large enough we have

[TABLE]

Proof.

Let $m_{0}$ be the positive integer such that $r^{m_{0}-1}\leq T<r^{m_{0}}$. By observing that

[TABLE]

we conclude that

[TABLE]

The first of the two terms is, by exploiting the arguments of the proof of Lemma 4, seen to be $\ll\,N^{1/2}T^{-\eta/2}$. Furthermore, the second term is, due to Lemma 4, up to a constant at most

[TABLE]

Hence the result of the Lemma is shown. ∎

In what follows $H,P$ and $T$ denote positive integers. We set

[TABLE]

Similar to inequality $(2.6)$ of [20] we can write

[TABLE]

Lemma 6**.**

Let $0<\kappa<1$ and assume the integers $P,H,T$ are such that

[TABLE]

Then for any $\delta>0$ we have

[TABLE]

and

[TABLE]

where $C_{0}=\frac{1}{2}+2(q^{1/2}-1)^{-1}$ is an absolute constant.

Proof.

We shall employ the inequality

[TABLE]

which is valid for all numbers with $|z|<z_{0}(\delta).$ Since

[TABLE]

we can apply (12) to obtain

[TABLE]

Observe that

[TABLE]

is a sum of trigonometric terms of frequencies at least $n_{2mH}\geq q^{2mH}$ in absolute value. Write

[TABLE]

where

[TABLE]

is the sum of trigonometric terms appearing in $U_{2m}^{2}(x)$ with frequencies at least $n_{2mH}$, and

[TABLE]

is the sum of the remaining terms in $U_{2m}^{2}(x)$, which have frequencies strictly less than $n_{2mH}$. It is shown in [26] and [20, p.246] that

[TABLE]

Hence

[TABLE]

where we define the integrand to be

[TABLE]

If we look at the $s$-th term in the above expansion, every factor

[TABLE]

is a sum of trigonometric terms, which have frequencies lying between $n_{2m_{i}H}$ and $2Tn_{(2m_{i}+1)H}$ in absolute value. The number of these terms is at most $3T^{2}H^{2}$. Thus any product

[TABLE]

is a sum of at most $3^{s}T^{2s}H^{2s}$ trigonometric terms, each of them being multiplied by a coefficient at most $\kappa^{s}$ and having frequency which is at least

[TABLE]

where we used the fact that

[TABLE]

We deduce that

[TABLE]

and

[TABLE]

Thus

[TABLE]

The second inequality of the lemma follows in precisely the same way. ∎

Proposition 7**.**

Let $M_{0}\geq 0,\,M\geq 1$ be positive integers and let $R\geq 1$ be a real number. Assume $f$ satisfies (8) and $\|f\|\geq M^{-3/5}$. Consider the set

[TABLE]

where $C_{0}>0$ is the constant from Lemma 6. Then

[TABLE]

Proof.

Without loss of generality, we may assume that $M_{0}=0$. We put $H=[M^{1/30}],\,T=M^{\lceil 4/\eta\rceil}$ and set

[TABLE]

We choose a positive integer $P$ such that

[TABLE]

Observe that $A\subseteq A_{1}\cup A_{2}$, where

[TABLE]

We are going to give estimates for the measure of these sets using the Chebyshev–Markov inequality. In order to do that, we observe that

[TABLE]

Also since $\eqref{kprel}$ is satisfied, we can apply Lemma 6. Regarding $A_{1}$, we estimate

[TABLE]

while for $A_{2}$, the Chebyshev–Markov inequality again gives

[TABLE]

∎

2.1.2 Proof of the upper bound

Let $N\geq 1$ be a positive integer sufficiently large. We set

[TABLE]

We define the functions $(\phi_{h}^{(j)})_{h\leq H_{1}}^{j\leq 2^{h}}$ as in [20]. Under this notation, inequality $(3.2)$ in [20] states that for each $0\leq\alpha<1$ there exists some index $j=j(\alpha)\leq 2^{h}$ such that

[TABLE]

For $1\leq h\leq H_{1},\,1\leq j\leq 2^{h},N\geq 1,M\geq 0$ set

[TABLE]

The following is a variation of Lemma $4$ in [20]. The proof relies on a method of Gal $\&$ Gal, see [13, Lemma $3.10$].

Lemma 8**.**

Let $n$ be the positive integer such that $2^{n}\leq N<2^{n+1}.$ There exist integers $(m_{l})_{l=1}^{n}$ such that $0\leq m_{l}<2^{n-l},\,1\leq l\leq n$ and

[TABLE]

In what follows we set

[TABLE]

Define the sets

[TABLE]

Lemma 9**.**

Let $0<\delta_{0}<1$. There exists $n_{0}=n_{0}(\delta_{0})\in\mathbb{N}$ such that

[TABLE]

Proof.

By (19) we have

[TABLE]

Applying Proposition 7 with $M_{0}=0,M=2^{n}$, $R=1$ and $f=\phi_{h}^{(j)}-\int\phi_{h}^{(j)}(t)\mathrm{d}t$, we get

[TABLE]

hence

[TABLE]

Applying Proposition 7 with $M_{0}=2^{n}+m2^{l},\,M=2^{l-1},R=2^{(n-l)/3}$ and $f=\phi_{h}^{(j)}-\int\phi_{h}^{(j)}(t)\mathrm{d}t$ we obtain

[TABLE]

We now deduce that

[TABLE]

The conclusion of the Lemma is now evident. ∎

We may now proceed to the final part of the proof of the upper bound. Choose an arbitrary $0\leq\alpha<1$. By (20) we obtain

[TABLE]

for all $0\leq x<1$ lying outside a set of $\mu$–measure at most $\delta_{0}$. Hence for those $0\leq x<1$ we obtain for any $0\leq\alpha<\beta<1$

[TABLE]

Taking the supremum over all $0\leq\alpha<\beta<1$, we get

[TABLE]

for all $x$ in a set of $\mu$–measure at most $\delta_{0}$. Now letting $\delta\to 0$ and then $\delta_{0}\to 0$ we obtain the requested upper bound in Theorem 1.

Remark 1**.**

It is worth pointing out a minor oversight in Philipp’s original proof [20]. The reader can easily check that a correct application of Philipp’s Proposition [20, p.244] to estimate $\lambda(H_{n})$ leads to a bound of the form $\lambda(H(n,j,h,l,m))\ll 2^{7n/24},$ hence relation $(3.8)$ in [20] is wrong. The problem can be overcome if we change the definition of the functions $g(x)$ in $(2.5)$ to $g(x)=\sum_{1\leq j\leq T}c_{j}\cos(2\pi jx),$ with $T$ a sufficiently large integer. It is likely that this oversight is due to the fact that the symbol $N$ is used for both $D_{N}(n_{k}x)$ and $N=2^{l-1}$ in the application of Philipp’s Proposition.

2.2 The lower bound

Given a sequence $(n_{k})_{k=1}^{\infty}$, Koksma’s Inequality implies that

[TABLE]

see [18, p.143] for more details. Thus the lower bound in Theorem 1 will follow immediately if we prove the following partial generalisation of the result of Erdős and Gal in [6]:

Proposition 10**.**

Let $(n_{k})_{k=1}^{\infty}$ be a lacunary sequence of integers such that (1) is satisfied. If $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\ll|t|^{-\eta},|t|\to\infty$ then

[TABLE]

The proof of Proposition 10 is essentially the same as in [6], with the only modifications being those relevant to the fact that $\mu$ is a probability measure other than the Lebesgue measure. We present here all steps of the proof which are essentially different and refer the reader to [6] for the remaining parts.

2.2.1 On the number of solutions of certain Diophantine inequalities

In what follows the postive integers $p,N$ are fixed, the sequence $(n_{k})_{k=1}^{\infty}$ is as in Theorem 1 and

[TABLE]

is a linear form in $2p$ variables which are allowed to take values in the set $\{n_{1},\ldots,n_{N}\}.$

Lemma 11**.**

For any $0<\alpha<\beta,$

[TABLE]

Proof.

This is Lemma $1$ in [6]. ∎

In what follows, given $s\in\mathbb{R}$ and $r>0$ we write $B(s,r):=(s-r,s+r)$ for the interval with center $s$ and length $2r$.

Lemma 12**.**

For positive integers $N,K\geq 1$ and $s\in\mathbb{Z}$ we write $\phi(1,N;s,K)$ for the number of pairs $(n_{k},n_{l})$ with $1\leq k,l\leq N$ such that $n_{k}-n_{l}\in B(s,2^{K-1})$ and $n_{k}\neq n_{l}$. Then

[TABLE]

Proof.

Since $\phi(1,N;s,K)=\phi(1,N;-s,K),$ we may assume without loss of generality that $s\geq 0$. If $0\leq s\leq 2^{K-1}$ then $\frac{1}{2}\phi(1,N;s,K)$ is at most equal to the number of pairs $(n_{k},n_{l})$ such that $1\leq n_{k}-n_{l}\leq 2^{K}$. If $(n_{k},n_{l})$ is such a pair, then

[TABLE]

hence $1\leq n_{k}\leq 2^{K}(1-q^{-1})^{-1}$ and by Lemma 11 the number of admissible $n_{k}$’s is at most $\dfrac{\log(2^{K}q^{2}(q-1)^{-1})}{\log q}\cdot$ Now we fix an admissible value of $n_{k}$ and we count the number of $n_{l}$’s which are acceptable for that specific $n_{k}$. Such $n_{l}$’s satisfy $1\leq n_{l}<n_{k}\leq 2^{K}(1-q^{-1})^{-1}$, so by Lemma 11 their number is at most $\dfrac{\log(2^{K}q^{2}(q-1)^{-1})}{\log q}$. Hence the number of possible pairs $(n_{k},n_{l})$ is at most

[TABLE]

On the other hand, if $s>2^{K-1}$ then $s+2^{K-1}\geq n_{k}-n_{l}\geq n_{k}(1-q^{-1})$ and $n_{k}\geq\max(1,s-2^{K-1})=s-2^{K-1}$, hence

[TABLE]

and by Lemma 11 there are at most

[TABLE]

possible values for $n_{k}$. Regarding $n_{l}$, we get $n_{k}-s-2^{K-1}\leq n_{l}\leq n_{k}-s+2^{K-1}$. Considering the cases $n_{k}\leq 2^{K-1}$ and $n_{k}>2^{K-1}$ separately, Lemma 11 gives at most $\dfrac{\log(2^{K}q)}{\log q}$ values for $n_{l}$, and the number of pairs $(n_{k},n_{l})$ is again bounded above by

[TABLE]

∎

Lemma 13**.**

For positive integers $N,K\geq 1$ and $s\in\mathbb{Z}$ let $\phi_{p}(s,K)$ be the number of pairs $(x_{p},y_{p})$ such that $A(x,y)\in B(s,2^{K-1}),$ subject to the restrictions $x_{1}\leq\ldots\leq x_{p}$ and $y_{1}\leq\ldots\leq y_{p}$. Then

[TABLE]

Proof.

Since $\phi_{p}(s,K)=\phi_{p}(-s,K)$, we may assume without loss of generality that $s\geq 0$. First we count the number of requested pairs $(x_{p},y_{p})$ for which $x_{p}\leq 2s+2^{K}$. The assumptions imply that

[TABLE]

hence $p^{-1}(s-2^{K-1})\leq x_{p}\leq 2^{K+2}$. When $s>3\cdot 2^{K-1}$, we have $\frac{1}{p}(s-2^{K-1})\leq x_{p}\leq 4(s-2^{K-1})$ and by Lemma 11 there are at most

[TABLE]

possible values for $x_{p}$. When $0\leq s\leq 3\cdot 2^{K-1},$ we have $1\leq x_{p}\leq 2s+2^{K}\leq 2^{K+2}$ and there are at most

[TABLE]

values of $x_{p}$. In both cases for $s$, there are at most $N$ choices for $y_{p}$, so the number of possible pairs $(x_{p},y_{p})$ with $x_{p}\leq 2^{K+2}$ is bounded above by

[TABLE]

Next we count the number of requested pairs $(x_{p},y_{p})$ for which $x_{p}>2^{K+2}$. The assumptions now imply that

[TABLE]

hence

[TABLE]

and the number of possible values for $y_{p}$ is by Lemma 11 at most $\dfrac{\log(4p^{2}q)}{\log q}$. Since there are at most $N$ possible choices for $x_{p}$, we have the upper bound

[TABLE]

for the number of pairs $(x_{p},y_{p})$ with $x_{p}>2s+2^{K-1}.$ Combining the estimates for the two cases we obtain the requested bound (24).

∎

Lemma 14**.**

For $1\leq p\leq N$ and $K\geq 1$ we write $\phi(p,N)$ for the number of solutions of $A(x,y)=0$ and $\phi(p,N;s,K)$ for the number of solutions of $A(x,y)\in B(s,2^{K-1})$, both subject to the restrictions $x_{1}\leq\ldots\leq x_{p}$ and $y_{1}\leq\ldots\leq y_{p}$. Then

[TABLE]

and

[TABLE]

Proof.

Inequality (25) is proved in Lemma $7$ of [6]. In order to prove (26), we fix the value of $K$ and use induction on $p\geq 1$. For $p=1$, (26) is implied by (23). Now we assume (26) is true for $1,2,\ldots,p-1$ and we seek an upper estimate for $\phi(p,N;s,K)$. To do this, we consider separately two sets of solutions: First, those $2p$-tuples $(x_{1},\ldots,y_{p})$ with $x_{1}=y_{1}$. Then the number of tuples $(x_{2},\ldots,x_{p},y_{2},\ldots,y_{p})$ with $s-2^{K-1}\leq(x_{2}+\ldots+x_{p})-(y_{2}+\ldots+y_{p})\leq s+2^{K-1}$ is at most $\phi(p-1,N;s,K)$ and there are $N$ possible values for $(x_{1},y_{1})$, hence we have at most

[TABLE]

solutions of that kind. Next we consider $2p$-tuples with $x_{1}\neq y_{1}$. By (24) the number of $2(p-1)$-tuples $(x_{2},\ldots,x_{p},y_{2},\ldots,y_{p})$ is at most

[TABLE]

For each such $2(p-1)$-tuple, the number of acceptable pairs $(x_{1},y_{1})$ with $x_{1}\neq y_{1}$ is given by (23) and is at most

[TABLE]

Combining the two cases, we obtain

[TABLE]

∎

Now we are able to prove the final goal of this subsection, which is giving an estimate for the number of solutions of equations of the form $A(x,y)\in B(s,2^{K-1})$. The result follows immediately from Lemma 14 since each solution to the aforementioned equation under the restrictions $x_{1}\leq\ldots x_{p},y_{1}\leq\ldots\leq y_{p}$ gives rise to $(p!)^{2}$ solutions $(x_{1},\ldots,y_{p})$.

Lemma 15**.**

For positive integers $1\leq p\leq N$, there exists a constant $c>0$ such that for any $s\in\mathbb{Z}$

[TABLE]

and

[TABLE]

2.2.2 Metrical Estimates on Exponential Sums

The previous Lemmas on the number of solutions of Diophantine equations with linear forms are used to estimate the moments of the function

[TABLE]

For $p\geq 1$ and $0\leq\alpha<\beta\leq 1$ we need to estimate the integral

[TABLE]

This in turn is used to provide estimates for the function

[TABLE]

The following lemma shows that if a probability measure on $[0,1)$ has Fourier transform with polynomial decay rate, then the same is true for any restriction of this measure to some subinterval.

Lemma 16**.**

Let $\mu$ be a probability measure on $[0,1)$ and $B=(\alpha,\beta)\subseteq[0,1)$ be a subinterval with $\mu(B)>0$. Let $\nu$ be the probability measure defined by $\nu(A)=\frac{1}{\mu(B)}\mu(A\cap B)$ for any subset $A\subseteq[0,1)$. If (4) holds for some $\eta>0$, then

[TABLE]

We include the proof of the Lemma in the Appendix at the end of the paper.

Proposition 17**.**

If $\alpha,\beta$ are such that $\mu((\alpha,\beta))\geq 1/n_{1}^{\eta}\sqrt{N}$ and $1\leq p\leq 3\log\log N$, then

[TABLE]

for all $N$ large enough.

Proof.

By definition of $F(N;x)$ we have

[TABLE]

where the implicit constant in the $O$-estimate is equal to $1$. The first term is estimated by (27), the second term again by (27) and the trivial bound $|e(x)|\leq 1$, while for the third term we use (30) and (28). Thus

[TABLE]

where $c_{1}=\log(4q^{2}(q-1)^{-1})$ and $c_{2}>0$ is some constant depending only on $\eta$ and $q$. Using the inequality

[TABLE]

finally yields the requested estimate (31). ∎

Armed with Proposition 17 the analogue of [6, Lemma 8] follows. The proof is omitted, as it involves precisely the same arguments.

Lemma 18**.**

Let $\phi$ be the function defined in (29) and $0<\varepsilon<1$. Then

[TABLE]

The remaining steps for the proof of the lower bound in Theorem 1 go along the lines of [6, p. 77-80], with the appropriate modifications for the measure $\mu$ instead of the Lebesgue measure.

3 Proof of Theorem 2

We utilize the discrepancy estimate coming from Theorem 1 in order to improve the result in [14]. We set

[TABLE]

For any $i\geq 1$ let $(q_{n}^{(i)})_{n=1}^{\infty}$ be the sequence of denominators associated with the continued fraction expansion of $\alpha_{i},$ and set

[TABLE]

The proof of the theorem will be complete as long as we show that the set

[TABLE]

has Hausdorff dimension $\dim G=1$. To that end, let $\mu=\mu(N,\delta,\varepsilon)$ be any of the probability measures as in Theorem of Kaufman and Queffélec-Ramaré. Now for each real $\gamma$ define the sequence of indices

[TABLE]

Claim: The sequence $(N_{k}^{\gamma})_{k=1}^{\infty}$ is well-defined.

Proof of Claim: Since $\alpha_{i}$ is badly approximable, the sequence $(q_{n}^{(i)})_{n=1}^{\infty}$ is lacunary and also

[TABLE]

(see [22, p.288, 297] for more details). Hence the lacunarity property together with Theorem 1 imply that for finally all $N\geq 1$ we have

[TABLE]

Here the constant $C_{i}>0$ depends on $\alpha_{i}\in{\rm{\bf Bad}}$ as in Theorem 1. For these values of $\beta$ lying in a set of full $\mu$-measure, the definition of discrepancy yields

[TABLE]

for all $\gamma\in\mathbb{R}$. Hence

[TABLE]

for all $N$ sufficiently large. This inequality shows that for all $k\geq 1$ there exists $N_{k}^{\gamma}\in\mathbb{N}$ such that

[TABLE]

The claim is proved.

Thus for all $k=1,2,\ldots$ and for all $\beta$ in a set of full $\mu$–measure we have

[TABLE]

Thus $\mu(G_{i})=1$ for all $i=1,2,\ldots$ and hence $\mu(G)=1$. Since $\mu=\mu(N,\delta,\varepsilon)$ was arbitrarily chosen, by property (ii) of the theorem of Kaufman and Queffélec–Ramaré together with the mass distribution principle. cf. [5, p. 975], we conclude that

[TABLE]

so $\dim G=1$ as required.

4 Appendix : The decay rate of the Fourier transform of the restricted measure.

Here we present a proof of Lemma 16, since we have not been able to locate the statement in the literature. The proof follows the one of an analogous result in [16, p. 252].

Since (4) holds, there exists a constant $C_{1}>0$ such that

[TABLE]

Let $\phi:\mathbb{R}\rightarrow(0,\infty)$ be a $C^{\infty}$ function which is equal to $1$ on the interval $B=(\alpha,\beta)$. Since $\phi$ is $C^{\infty}$, we have

[TABLE]

where the convergence is uniform for all $x\in\mathbb{R}$. Furthermore, since $\phi$ is a $C^{\infty}$ function, there exists a constant $C_{\eta}>0$ such that

[TABLE]

Set $S=\sum\limits_{k=-\infty}^{+\infty}(1+|k|)^{-(1+\eta)}<\infty.$ For the probability measure $\nu$ defined as in Lemma 16 we have for $t>0$

[TABLE]

We deal with the first of the two terms. The condition of summation implies that $\frac{1}{2}|t|\leq|t-k|\leq\frac{3}{2}|t|$. Hence employing (4) and (33) we get

[TABLE]

Regarding the second term, using the trivial bound $|\widehat{\mu}(t)|\leq 1$ together with (33) we get

[TABLE]

Combining the two estimates, we obtain

[TABLE]

with $C=2^{\eta}C_{\eta}(1+S)>0$. The same bound is also true for all real values of $t$ in view of the relation $|\widehat{\nu}(t)|=|\widehat{\nu}(-t)|$, hence the Lemma is proved.

**Acknowledgements: **We would like to thank Professor C. Aistleitner and Professor S. Velani for suggesting this direction of research and for many useful comments during the preparation of this paper. We also thank Bence Borda for pointing out Corollary 3 to us.