# Inhomogeneous Diophantine Approximation on $M_{0}$-sets

Volodymyr Pavlenkov111Research is supported by the British Academy in
partnership with the Academy of Medical Sciences, the Royal Academy of
Engineering, the Royal Society and Cara under the _Researchers at Risk
Fellowships_ Award RAR\100132
University of York
Igor Sikorsky Kyiv Polytechnic Institute  Evgeniy Zorin
University of York

###### Abstract

We prove new quantitative Schmidt-type theorem for Diophantine approximations
with restraint denominators on fractals (more precisely, on $M_{0}$-sets). Our
theorems introduce a sharp balance condition between the growth rate of the
sequence of denominators and the decay rate of the Fourier transform of a
Rajchman measure. Among the other things, this allows applications to
sequences of denominators of polynomial growth. In particular, we infer new
inhomogeneous Khintchine-Järnik type theorems with restraint denominators for
a broad family of denominator sequences. Furthermore, our results provide non-
trivial lower bounds for Hausdorff dimensions of intersections of two sets of
inhomogeneously well-approximable numbers with restraint denominators.

##  1 Introduction

Metric theory of Diophantine approximations originates in the classical
problem of approximating irrational numbers, such as $\sqrt{2}$ or $\pi$, by
fractions. It was known already in ancient times that some constants arising
in practice admit good rational approximations, providing nice precision while
having a relatively not so large denominator (for example, rational
approximations to $\pi$ such as $22/7\approx 3.14\dots$ and $355/113\approx
3.141593\dots$). A natural question to ask is to what extent such good
rational approximations are typical for real numbers? In other terms, for a
generic real number, what one should expect to have as a balance between the
precision of a rational approximation and the size of its denominators? A bit
surprisingly, almost all rational numbers allow essentially the same
asymptotic order of the best rational approximations when compared to the size
of denominators of these approximations. This fact was established by
Khintchine in 1929 (see Theorem KS in Subsection 2 below). Later, Schmidt
refined this result in 1964 by proving that, for almost all numbers, the best
rational approximations appear at quite regular intervals.

The development of metric theory of Diophantine approximations conducted to
research on more complicated situations. For example, studies of simultaneous
applications led to considering approximations of one real number by
convergents (the best rational approximations) of another one, which naturally
led to studies of approximations with constraints on available denominators.
At the same time, in another direction of research, the metric theory of
Diophantine approximations was extended to the context of fractal sets.
Motivation for the latter was manifold, ranging from pure extension of
existing theory to the demand by practical applications. Importantly, many of
sets of interest arising in the metric theory of Diophantine approximations
are fractals in nature (e.g. the sets of badly approximable numbers, well
approximable numbers, the sets of exact order etc.), so the studies of
Diophantine approximations on fractals provide, among the other things, a
potent tool for investigating intersections of such sets.

So, a natural progress in this field has led to a large stream of research on
metric theory of Diophantine approximations on fractals. It would have been an
impossible task to give a comprehensive literature review on this subject
within the framework of this paper, so we benefit from the possibility to
refer the reader to [3] for a general review of the subject and to [1] for
some examples of applications.

This paper develops further metric theory of Diophantine approximations on so-
called $M_{0}$-sets.

######  Definition 1.

The set $F$ is called an $M_{0}$-set if it supports a non-atomic probability
Borel measure $\mu$ ($\text{supp}(\mu)\subset F$) whose Fourier transform
$\widehat{\mu}$ vanishes at infinity, i.e.

$\lim_{|t|\to\infty}|\widehat{\mu}(t)|=0,$

where

$\widehat{\mu}(t)\,:=\int e^{-2\pi itx}\,\mathrm{d}\mu(x),\hskip
28.45274pt(t\in\mathbb{Z})\,.\vspace{2mm}$

Such a measure $\mu$ is called a Rajchman measure.

The interest of studying $M_{0}$-sets is bifold. On the one hand, there are
quite many classical (fractal) sets of interest that support a Rajchman
measure (see, for example, [6, 7, 9, 12, 14, 19]). On another hand, it is
quite natural to expect, at least heuristically, that such measures’ behaviour
have significant similarities with Lebesgue measure: indeed, one would
naturally expect that, as $|\hat{\mu}(t)|$ tends to 0 as $t\to 0$, the
behaviour of $\mu$ is determined to a significant degree (whatever it means in
concrete terms) by some first coefficients, $\hat{\mu}(t)$ for $t=-N,\dots,N$.
And then, of course, the measure with Fourier transform
$\sum_{t=-N}^{N}\hat{\mu}(t)\exp\left(-2\pi it\right)$ is absolutely
continuous with respect to Lebesgue. See a more detailed discussion and some
further references in [17], where some quantitative results that illustrate
the heuristics above have been proven and some other results from the past
literature have been discussed from this point of view.

It is important to remark straight away that to have any meaningful
generalization of Khintchine Theorem to the case of sufficiently general
$\mu$, for instance to suitable Rajchman measures, we necessarily have to
consider approximations with restrained sequences of denominators. Indeed, a
classical construction by Kaufman [12], later updated by Queffelec-Ramaré
[19], shows that the set of badly approximable numbers supports a Rajchman
measure (indeed, an infinite family of such measures). So, it is well possible
that, with respect to certain Rajchman measures $\mu$, even with Fourier
transform quickly converging to 0, $\mu$-almost all numbers are badly
approximable, contrary to what we should have had with the usual Khintchine
Theorem.

In [17], Khintchine type theorem (in fact, Schmidt-type counting theorem) is
proven for sufficiently quickly growing sequences of denominators
$\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$. In fact, in [17] the convergence part
of Khintchine theorem is established in quite a general situation: for
arbitrary sequence of denominators $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ and
under one of the following two conditions on the measure $\mu$

$\sum_{n=1}^{\infty}\;\max_{k\in\mathbb{Z}/\\{0\\}}|\widehat{\mu}(kq_{n})|<\infty,$
(1)
$\sum_{n=1}^{\infty}\,\sum_{k\in\mathbb{Z}/\\{0\\}}\frac{|\widehat{\mu}(kq_{n})|}{|k|}<\infty.$
(2)

However, the most challenging divergence part is established only for Rajchman
measures with a sufficiently quickly decaying Fourier transform and, what is
the most relevant for the topic of this paper, under quite an important growth
condition on the sequence $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ (in
particular, the growth of $\mathcal{A}$ required there strictly exceeds the
polynomial one, and indeed should be of order of magnitude
$q_{n}\geq\exp\left(n^{c}\right)$ for some $c>0$).

In this paper, we prove a quantitative Schmidt-type theorem under a balance
condition between the growth rate of the sequence
$\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ and the decay rate of the Fourier
transform $\widehat{\mu}$. This balance condition allows, on the one hand, to
establish Khintchine Theorem for some well-known measures (for instance,
Kaufmann measures on the sets on badly- or well-approximable numbers) for the
sequences of denominators of $\mathcal{A}$ of polynomial growth. On another
hand, this balance condition allows applications even to Rajchman measures
with a relatively slowly decaying Fourier transform. For example, recently it
was proven that, under quite broad conditions, there exist Rajchman measures
on self-similar sets associated to contractions, but the decay rate of the
Fourier transforms of the measures constructed is, when $t\to\infty$,
$O(\log(|t|)^{-\delta})$, where $\delta>0$ is a constant not explicitly given
in the paper, and the methods implied suggest that $\delta>0$ is very small
(much smaller than one, for instance). In such situation, our main result,
Theorem 1, still allows to obtain a Khintchine type theorem for sufficiently
sparse sequences of denominators.

The plan of this paper is as follows.

In Section 2 we give some definitions and introduce some basic notation needed
in our further discussion, and, in particular, to state our main result. In
the same section we state Khintchine’s theorem and also explain Schmidt’s
result, both of them are mentioned in the introduction.

In Section 3 we state the main result of this paper.

In Section 4 we provide some examples of applications of our main result.

Finally, we give proofs in Section 5.

##  2 Khintchine-Szüsz $0$ and $1$ Law and Schmidt generalization

We begin by giving a few of basic definitions.

Given a real number $\gamma\in\mathbb{I}$, approximation function
$\psi:\mathbb{N}\rightarrow\mathbb{I}$ and a natural number $q\in\mathbb{N}$,
let

$E(q,\gamma,\psi):=\left\\{x\in\mathbb{I}:\|qx-\gamma\|\leq\psi(q)\right\\},$
(3)

where $\|\alpha\|:=\min\\{|\alpha-m|:m\in\mathbb{Z}\\}$ denotes the distance
from $\alpha\in\mathbb{R}$ to the nearest integer. For any sequence
$\mathcal{A}=(q_{n})_{n\in\mathbb{N}}\subset\mathbb{N}$ of natural numbers,
define, for any $N\in\mathbb{N}$, the counting function by

$R(x,N)\,=\,R(x,N;\gamma,\psi,\mathcal{A}):=\\#\big{\\{}1\leq n\leq N:x\in
E(q_{n},\gamma,\psi)\big{\\}}.$ (4)

Note that the definition above could be rewritten as

$R(x,N)=\sum_{n=1}^{N}\chi_{{}_{E(q_{n},\gamma,\psi)}}(x),$ (5)

where $\chi_{{}_{E(q_{n},\gamma,\psi)}}$ is the characteristic function of the
set $E(q_{n},\gamma,\psi).$

Recall that the set of inhomogeneous $\psi$-well approximable real numbers

$W_{\mathcal{A}}(\gamma;\psi):=\left\\{x\in\mathbb{I}:\|q_{n}x-\gamma\|\leq\psi(q_{n})\text{
for infinitely many }n\in\mathbb{N}\right\\}.\vspace{2mm}$ (6)

Note that

$W_{\mathcal{A}}(\gamma;\psi)=\limsup
E(q_{n},\gamma,\psi)=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}E(q_{k},\gamma,\psi).$
(7)

We gathered in Proposition 1 below some basic properties of the function
$R(x,N)$ and sets $W_{\mathcal{A}}(\gamma;\psi)$ that we will use in the proof
of our main result.

######  Proposition 1.

Let $x\in\mathbb{I},$ $\gamma\in\mathbb{I}$ and let $\psi,\psi_{1},\psi_{2}$
be auxiliary functions, that is
$\psi,\psi_{1},\psi_{2}:\mathbb{N}\rightarrow\mathbb{I}$. Let
$\mathcal{A}=(q_{n})_{n\in\mathbb{N}}\subset\mathbb{N}$ be a sequence of
natural numbers. Then,

_(i)_ $x\in W_{\mathcal{A}}(\gamma;\psi)$ if and only if

$\mathop{\lim}\limits_{N\to\infty}R(x,N)=\infty;$

_(ii)_ if $\psi_{1}(q_{n})\leq\psi_{2}(q_{n})$ for all $n\in\mathbb{N},$ then

$W_{\mathcal{A}}(\gamma;\psi_{1})\subset W_{\mathcal{A}}(\gamma;\psi_{2}),$

and

$R(x,N;\gamma,\psi_{1},\mathcal{A})\leq R(x,N;\gamma,\psi_{2},\mathcal{A});$

_(iii)_ if for some auxiliary functions $\psi_{1},\psi_{2}$ we have $x\notin
W_{\mathcal{A}}(\gamma;\psi_{2})$, then

$R(x,N;\gamma,\max\\{\psi_{1},\psi_{2}\\},\mathcal{A})=R(x,N;\gamma,\psi_{1},\mathcal{A})+O(1).$

###### Proof of Proposition 1.

Statements (i) and (ii) of Proposition 1 follow directly from definitions (3),
(4) and (6). In order to prove (iii), note that from (ii) of Proposition 1 we
have, for any $N\in\mathbb{N}$,

$R(x,N;\gamma,\psi_{1},\mathcal{A})\leq
R(x,N;\gamma,\max\\{\psi_{1},\psi_{2}\\},\mathcal{A})\leq
R(x,N;\gamma,\psi_{1},\mathcal{A})+R(x,N;\gamma,\psi_{2},\mathcal{A}).$ (8)

Since $x\notin W_{\mathcal{A}}(\gamma;\psi_{2}),$ (i) of Proposition 1 implies
that $R(x,N;\gamma,\psi_{2},\mathcal{A})$ remains bounded as $N\to\infty.$
This fact together with (8) completes the proof of Proposition 1. ∎

Khintchine-Szüsz Theorem provides the $0$ and $1$ law for the Lebesgue size of
the set $W_{\mathbb{N}}(\gamma;\psi)$. Khintchine [13] proved the homogeneous
statement (the case $\gamma=0$) in 1924, and later, in 1954, Szüsz [21]
generalized Khintchine’s result to the inhomogeneous case.

######  Theorem  KS .

Let $\gamma\in\mathbb{I}$ and $\psi:\mathbb{N}\rightarrow\mathbb{I}$ be a
real, positive non-increasing function. Then

$m\big{(}W_{\mathbb{N}}(\gamma;\psi)\big{)}=\left\\{\begin{array}[]{ll}0&\mbox{if}\;\;\;\sum\limits_{n=1}^{\infty}\psi(n)<\infty\;,\\\\[8.61108pt]
1&\mbox{if}\;\;\;\sum\limits_{n=1}^{\infty}\psi(n)=\infty\;,\end{array}\right.$

where $m$ is the Lebesgue measure.

In 1964 Schmidt [20] generalized Khintchine-Szüsz Theorem giving a
quantitative result on the size of counting function $R(x,N)$ given by (4)
with decreasing auxiliary function $\psi$ and $\mathcal{A}=\mathbb{N}$:

$R(x,N)\ =\
2\Psi(N)+O\Big{(}\Psi(N)^{1/2}\left(\log(\Psi(N))\right)^{2+\varepsilon}\Big{)},\quad
N\in\mathbb{N},$

for every $\varepsilon>0$ and for $m$-almost all $x,$ where

$\Psi(N):=\sum_{n=1}^{N}\psi(n).$

Our main result, given in Section 3 below, is a Schmidt-type counting theorem.

##  3 Main result

The statement of our main theorem uses the following definition.

######  Definition 2.

Let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an increasing sequence of
natural numbers and let $\alpha\in(0,1)$ be a real number. We say that
$\mathcal{A}$ is _$\alpha$ -separated_ if there exists $m_{0}\in\mathbb{N}$
such that, for all $m,n\in\mathbb{N}$, $m_{0}\leq m<n$, the set of solutions
$(s,t)\in\mathbb{N}^{2}$ of the following system of Diophantine inequalities

$\begin{cases}1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha},\\\ s\leq m^{5},\end{cases}$

is empty.

######  Remark 1.

Note that Definition 2 is similar, but not identical, to the one in [17] (see
[17, p. 12]). In Definition 2 we have optimized one exponent, so we have
$m^{5}$ in place of $m^{12}$ in [17]. As a result, if a sequence of
denominators $\mathcal{A}$ is $\alpha$-separated in the sense of [17], than it
is necessarily $\alpha$-separated in the sense of Definition 2 (but not
necessarily vice versa).

######  Theorem 1.

Let $\mu$ be a non-atomic probability measure supported on a subset $F$ of $\
\mathbb{I}\,$. Let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an
$\alpha$-separated increasing sequence of natural numbers for some
$\alpha\in(0,1).$ Suppose there exists a real constant $\rho>2$ and a
monotonically decreasing function $h:\mathbb{N}\to\mathbb{I}$ verifying

$h(q_{n})\,=\,O\ (n^{-\rho}),\quad n\in\mathbb{N}.$ (9)

such that

$|\widehat{\mu}(t)|\,=\,O\ (h(|t|)),\quad t\in\mathbb{Z}.$ (10)

Then, for all given $\gamma\in\mathbb{I},$
$\psi:\mathbb{N}\rightarrow\mathbb{I}$ and for any $\varepsilon>0$ the
counting function $R(x,N)$ satisfies

$\begin{array}[]{ll}R(x,N)\ =\
2\Psi(N)+O\Big{(}\left(\Psi(N)+E(N)\right)^{1/2}\left(\log(\Psi(N)+E(N)+2\right)\big{)}^{2+\varepsilon}\Big{)}\end{array}$
(11)

for $\mu$-almost all $x\in F$, where

$\Psi(N):=\sum_{n=1}^{N}\psi(q_{n})\,$ (12)

and

$E(N)\ :=\ \mathop{\sum\sum}_{1\leq m<n\leq N}(q_{m},q_{n})\
\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\,,$ (13)

here $(q_{m},q_{n})$ is the gcd of natural numbers $q_{m}$ and $q_{n}.$

######  Remark 2.

If, under all conditions of Theorem 1, for some $\varepsilon>0$ the gcd term
$E(N),$ given by (13), satisfies equality

$E(N)=O\left(\Psi^{2-\varepsilon}(N)\right),\quad N\in\mathbb{N},$ (14)

and $\Psi(N)\to\infty$ as $N\to\infty,$ then for the counting function
$R(x,N)$ we have the following asymptotic statement

$\lim_{N\to\infty}\frac{R(x,N)}{2\Psi(N)}=1,$ (15)

for $\mu$-almost all $x\in F.$

We postpone the proof of Theorem 1 until Section 5. In the following Section 4
we give some examples of applications of this theorem.

##  4 Applications of main result

In this section we give some examples of application of Theorem 1.

###  4.1 Example of sequence $\mathcal{A}$ with polynomial growth and fitting
all conditions of Theorem 1

In this part, we construct, as an example, a family of sequences of
denominators $\mathcal{A}$ of polynomial growth for which Theorem 1 gives non-
trivial results.

######  Example 1.

In this example, we provide a construction of an $\alpha$-separated sequence
$(q_{m})_{m\in\mathbb{N}}$ of polynomial growth (more precisely, verifying
(19) for parameters $\rho_{1}<\rho_{2}$ as below).

Let

$1<\rho_{1}<6\rho_{1}<\rho_{2}$ (16)

and $c>1$ be real parameters. Let us choose a sequence of integers
$(n_{k})_{k\in\mathbb{N}}$ as follows. First, choose $n_{1}$ to be a
sufficiently large positive integer, so that
$n_{1}^{\rho_{1}}<n_{1}^{\rho_{2}}/c$. Then, for every $k\in\mathbb{N}$,
choose arbitrarily $n_{k+1}$ to be an integer in the range

$\lfloor n_{k}^{\rho_{2}}\rfloor/c\leq n_{k+1}\leq\lfloor
n_{k}^{\rho_{2}}\rfloor.$ (17)

For each $k\in\mathbb{N}$, define sets of integers

$\mathcal{Q}_{k}:=\left\\{s\cdot n_{k}\mid s=1,\dots,\lfloor
n_{k}^{\rho_{1}-1}\rfloor\right\\}.$ (18)

Then, define $\mathcal{A}=(q_{m})_{m\in\mathbb{N}}$ to be the set of numbers

$\bigcup_{k\in\mathbb{N}}\mathcal{Q}_{k}$

put in increasing order.

We claim that we have, for every $m\in\mathbb{N}$ sufficiently large (so that
we have $m>n_{1}^{\rho_{1}-1}$),

$\frac{\log q_{m}}{\log m}\leq\frac{\rho_{2}}{\rho_{1}-1}+1.$ (19)

Indeed, every $q_{m}$ has a form $s\cdot n_{k}$ for some $s,k\in\mathbb{N}$,
where $s\leq n_{k}^{\rho_{1}-1}$. The lower bound on $m$ above means that
$k\geq 2$. Then,

$m\geq s+\lfloor n_{k-1}^{\rho_{1}-1}\rfloor\geq s+\lfloor
n_{k}^{(\rho_{1}-1)/\rho_{2}}\rfloor.$

Consider two cases: the first one when $s\leq\lfloor
n_{k}^{(\rho_{1}-1)/\rho_{2}}\rfloor$, and the second one when $s>\lfloor
n_{k}^{(\rho_{1}-1)/\rho_{2}}\rfloor$.

Case 1. Assume $s\leq\lfloor n_{k}^{(\rho_{1}-1)/\rho_{2}}\rfloor$. Then, we
have $m\geq n_{k}^{(\rho_{1}-1)/\rho_{2}}$ (we use here $s\geq 1$), hence

$\frac{\log q_{m}}{\log m}\leq\frac{\log s+\log n_{k}}{\log
n_{k}^{(\rho_{1}-1)/\rho_{2}}}\leq\frac{\log
n_{k}^{(\rho_{1}-1)/\rho_{2}}+\log n_{k}}{\log
n_{k}^{(\rho_{1}-1)/\rho_{2}}}\leq\frac{\rho_{2}}{\rho_{1}-1}+1.$

This proves (19) in the first case.

Case 2. Assume $s>\lfloor n_{k}^{(\rho_{1}-1)/\rho_{2}}\rfloor$. As $s$ is an
integer, we also have then $s>n_{k}^{(\rho_{1}-1)/\rho_{2}}$, hence

$\frac{\log q_{m}}{\log m}\leq\frac{\log s+\log n_{k}}{\log s}\leq
1+\frac{\log n_{k}}{\log s}<\frac{\rho_{2}}{\rho_{1}-1}+1.$

So we have verified (19) in both cases.

Furthermore, let’s remark for a further use

$q_{m}>m.$ (20)

Indeed, it follows from our construction that, for every $m\in\mathbb{N}$,

$q_{m+1}-q_{m}>1,$

and the claim follows by induction.

We proceed with proving that

$\text{ the sequence }(q_{n})\text{ is }\alpha\text{-separated for
}\alpha=1/\rho_{1}.$ (21)

So, let $m,n,s\in\mathbb{N}$ verifying $m<n$ and $s<m^{5}$. In case if, for
some $k\in\mathbb{N}$, $q_{m},q_{n}\in\mathcal{Q}_{k}$, then $q_{m}$ and
$q_{n}$ are both divisible by $n_{k}$, so in case if, for some
$t\in\mathbb{N},$ $|sq_{m}-tq_{n}|\geq 1$, we necessarily have

$|sq_{m}-tq_{n}|\geq n_{k},$

hence

$|sq_{m}-tq_{n}|\geq q_{m}^{1/\rho_{1}}.$

To deal with the complimentary case, if $q_{m}\in\mathcal{Q}_{k}$,
$q_{n}\in\mathcal{Q}_{l}$ for some $k<l$, note first that it could be deduced
from the construction of the sequence $(q_{m})_{m\in\mathbb{N}}$, that
$q_{m}\geq m^{\rho_{1}}$. Then, we have

$|sq_{m}|<m^{5}q_{m}<q_{m}^{6}\leq n_{k}^{6\rho_{1}}.$

So, for $n_{k}$ sufficiently large (otherwise speaking, for $m$ sufficiently
large) and taking into account (16), we have

$|sq_{m}|<\lfloor n_{k}^{\rho_{2}}\rfloor/(2c)\leq n_{k+1}/2\leq q_{n}/2.$

Thus in this case we have that either $s=t=0$, in which case, of course,
$sq_{m}-tq_{n}=0$, or

$|sq_{m}-tq_{n}|\geq q_{m}.$

This complets the verification of (21).

######  Example 2.

We continue to work in the framework of Example 1, now assuming in addition
$c\geq 2$ in (17). We want to update the construction from that example to
ensure, apart from polynomial growth and $\alpha$-separation, that also the
term $E(N)$, defined in the statement of Theorem 1, has a size that allows
optimal result in Theorem 1. More precisely, we are going to rectify Example 1
above, by constructing a sequence
$\widetilde{\mathcal{A}}=(\widetilde{q}_{t})_{t\in\mathbb{N}}$, so that we
have

$E(N)=\ \mathop{\sum\sum}_{1\leq m<n\leq
N}(\widetilde{q}_{m},\widetilde{q}_{n})\
\min\left(\frac{\psi(\widetilde{q}_{m})}{\widetilde{q}_{m}},\frac{\psi(\widetilde{q}_{n})}{\widetilde{q}_{n}}\right)=O\left(\sum_{n=1}^{N}\psi(\widetilde{q}_{n})\right).$
(22)

Recall that we assume $c=2$. We choose all the numbers in the sequence
$(n_{k})_{k\in\mathbb{N}}$ to be prime, this is possible by Bertrand’s
postulate.

Next, we modify sets $\mathcal{Q}_{k}$, defined in (23), as follows:

$\widetilde{\mathcal{Q}}_{k}:=\left\\{s\cdot n_{k}\mid s=1,\dots,\lfloor
n_{k}^{\rho_{1}-1}\rfloor,\;s\text{ is prime}\rfloor\right\\}.$ (23)

Then, similarly to Example 1, we define
$\widetilde{\mathcal{A}}=(\widetilde{q}_{t})_{t\in\mathbb{N}}$ to be the set
of numbers

$\bigcup_{k\in\mathbb{N}}\widetilde{\mathcal{Q}}_{k}$

put in increasing order.

For the sake of comparison, let us denote by $\mathcal{A}$ the set of
denominators built as in Example 1 using the same sequence of primes
$(n_{k})_{k\in\mathbb{N}}$ that we have just constructed for
$\widetilde{\mathcal{A}}$. Then, it follows from the definitions that the set
of values of $\widetilde{\mathcal{A}}$ is the subset of values of
$\mathcal{A}$, that is, for every index $t\in\mathbb{N}$ there exists an index
$m\in\mathbb{N}$ such that $\widetilde{q}_{t}=q_{m}$. It follows then from the
law of distribution of prime numbers that, for $t\in\mathbb{N}$ large enough,
$t\asymp\frac{m}{\log m}$, hence, for every $\varepsilon>0$, we have, for all
indices $t\in\mathbb{N}$ large enough,

$\frac{\log\widetilde{q}_{t}}{\log
t}\leq\frac{\rho_{2}}{\rho_{1}-1}+1+\varepsilon,$

so $\left(\widetilde{q}_{t}\right)_{t\in\mathbb{N}}$ has a polynomial growth.

We proceed with establishing (22). We claim that, for every $n\in\mathbb{N}$,
we have

$\sum_{m=1}^{n-1}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}=O(1),$
(24)

which clearly implies (22).

By construction, $\widetilde{q}_{n}=s\cdot n_{k}$ for some $k\in\mathbb{N}$
and prime $s$ verifying $s\leq n_{k}^{\rho_{1}-1}$. Similarly, for all
$m\in\mathbb{N}$, $\widetilde{q}_{m}=s_{1}\cdot n_{l}$. Let us denote by
$I_{1}$ the collection of indices $m<n$ such that $l=k$, and $I_{2}$ the
complimentary set of indices $m<n$ such that $l<k$. Naturally,
$\\{1,\dots,n-1\\}=I_{1}\sqcup I_{2}$, hence

$\sum_{m=1}^{n}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}=\sum_{m\in
I_{1}}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}+\sum_{m\in
I_{2}}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}.$

We are going to show that both sums in the right-hand side are $O(1)$, which
is equivalent to proving (24).

First, we consider the sum $\sum_{m\in I_{1}}$. In this case, by definition of
$I_{1}$, $k=l$. Then, we necessarily have $s_{1}<s$, because $m<n$. Moreover,
we have in this case

$(q_{m},q_{n})=(s_{1}n_{k},sn_{k})=n_{k},$

because $s_{1}$ and $s$ are two distinct primes. So,

$\sum_{m\in
I_{1}}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}=\sum_{\begin{subarray}{c}2\leq
s_{1}<s\\\ s_{1}\text{ prime}\end{subarray}}\frac{1}{s}<1.$

Next, we consider the sum $\sum_{m\in I_{2}}$. In this case, we have $l<k$,
hence, by construction of sequences $(n_{t})_{t\in\mathbb{N}}$ and
$\widetilde{\mathcal{A}}$, we necessarily have $\widetilde{q}_{m}\leq
n_{k-1}^{\rho_{1}}\leq n_{k}^{\rho_{1}/\rho_{2}}$, hence
$\widetilde{q}_{n}\geq n_{k}>\widetilde{q}_{m}^{\rho_{2}/\rho_{1}}$. We find
thus

$\sum_{m\in
I_{2}}\frac{(\widetilde{q}_{m},\widetilde{q}_{n})}{\widetilde{q}_{n}}\leq\sum_{m}\frac{\widetilde{q}_{m}}{\widetilde{q}_{m}^{\rho_{2}/\rho_{1}}}<+\infty,$

where at the last step we use (16). This proves (24), hence (22).

###  4.2 Khintchine Theorem on the set of Liouville numbers

In this section, we establish the Khintchine-Szüsz theorem (for restraint
denominators) on the set of Liouville numbers:

$\mathbb{L}=\left\\{x\in\mathbb{R}\setminus\mathbb{Q}\;\;|\;\;\forall\;n\in\mathbb{N}\;\;\exists\;\;q\in\mathbb{N}:\|qx\|<q^{-n}\right\\}.$

In 2000 Bluhm [5] constructed a Rajchman measure $\mu_{\mathbb{L}}$ supported
on $\mathbb{L}$ showing that the set of Liouville numbers is an $M_{0}$ set.
In 2002 Bugeaud [8] calculated the exact decay rate of the Fourier transform
$\widehat{\mu_{\mathbb{L}}}:$

$|\widehat{\mu_{\mathbb{L}}}(t)|\leq\exp\left\\{-c_{2}\sqrt{\log(1+|t|)}\right\\},\quad
t\in\mathbb{Z},$ (25)

where $c_{2}$ is a positive absolute constant appearing in [8]. We will use
constant $c_{2}$ and measure $\widehat{\mu_{\mathbb{L}}}$ in the statement of
Theorem 2 below).

This allows us to infer the following theorem.

######  Theorem 2.

Let $\psi:\mathbb{N}\rightarrow\mathbb{I}$, $\gamma\in\mathbb{I}$,
$\alpha\in(0,1)$ and let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an
$\alpha$-separated increasing sequence of natural numbers with growth rate

$\log q_{n}\geq\left(\frac{\rho}{c_{2}}\log n\right)^{2},\quad n\geq n_{0},$
(26)

for some natural $n_{0}$, $\rho>2$ and $c_{2}$ from (25). Recall the measure
$\widehat{\mu_{\mathbb{L}}}$ described before the statement of the theorem and
appearing in (25).

Then, for any $\varepsilon>0$, we have the following asymptotic counting
result for $\widehat{\mu_{\mathbb{L}}}:$-almost every Liouville number $x$:

$\begin{array}[]{ll}R(x,N)\ =\
2\Psi(N)+O\Big{(}\left(\Psi(N)+E(N)\right)^{1/2}\left(\log(\Psi(N)+E(N)+2\right)\big{)}^{2+\varepsilon}\Big{)},\end{array}$

where

$\Psi(N):=\sum_{n=1}^{N}\psi(q_{n})\,$

and

$E(N)\ :=\ \mathop{\sum\sum}_{1\leq m<n\leq N}(q_{m},q_{n})\
\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right).$

###### Proof of Theorem 2.

We will deduce Theorem 2 from Theorem 1. In order to do this we put

$h(t)=\exp\left\\{-c_{2}\sqrt{\log(1+t)}\right\\},\quad t\in\mathbb{N}.$

Conditions (9) and (10) of Theorem 1 now follow from (25) and (26), therefore
all statements of Theorem 2 now follow from the corresponding statements of
Theorem 1. ∎

Consider the following two examples of sequences $\mathcal{A}$. We will use
these examples in Corollary 1 below.

  1. 1.

Let $\mathcal{S}$ be a finite set of $k$ distinct primes $p_{1},\ldots,p_{k}$.
It is shown in [17] that any sequece of $\mathcal{S}$-smooth numbers
$\mathcal{A}_{1}$ is an $\alpha$-separated sequence, for any $\alpha\in(0,1)$.
It is also shown there that $\mathcal{A}_{1}$ has a growth rate

$\log q_{n}>\frac{\log 2}{2}n^{\frac{1}{k}},\quad n\geq 2,$

and has the gcd error term $E(N)$ of order $E(N)=O(\Psi(N))$ (see Theorem 5 in
[17]).

  2. 2.

Let $\mathcal{A}_{2}$ be a subsequence of sequence $\widetilde{\mathcal{A}}$
verifying the growth condition (26). Since $\widetilde{\mathcal{A}}$ is an
$\alpha$-separated sequence and verifies (22), necessarily its subsequence
$\mathcal{A}_{2}$ is also $\alpha$-separated and verifies (22).

The following corollary is readily implied by Theorem 2 and properties of
sequences $\mathcal{A}_{1}$ and $\mathcal{A}_{2}.$

######  Corollary 1.

Let $\mathbb{L}$ be the set of Liouville numbers with Rajchman measure
$\mu_{\mathbb{L}}$ supported on it and verifying (25). Then, for any
$\gamma\in\mathbb{I}$ and constructed above sequences $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$, we have

$\mu_{\mathbb{L}}\big{(}W_{\mathcal{A}_{i}}(\gamma;\psi)\cap\mathbb{L}\big{)}=\left\\{\begin{array}[]{ll}0&\mbox{if}\;\;\;\sum\limits_{n=1}^{\infty}\psi(q_{n})<\infty\;,\\\\[8.61108pt]
1&\mbox{if}\;\;\;\sum\limits_{n=1}^{\infty}\psi(q_{n})=\infty\;,\quad\quad
i=1,2.\end{array}\right.$

###  4.3 Hausdorff and Fourier dimensions of sets
$W_{\mathcal{A}}(\gamma;\psi)$ and their intersections

In this section we deduce a nontrivial result on the Hausdorff and Fourier
dimensions of some subsets of the sets $W_{\mathcal{A}}(\gamma;\psi)$.

Recall that the Fourier dimension of a set $A\subset\mathbb{R}$ is a number
$\dim_{F}A,$ that is a supremum of all $\eta\in[0,1]$ such that

$|\widehat{\mu}(t)|=O\left(|t|^{-\frac{\eta}{2}}\right),\quad\text{as}\quad|t|\to\infty,$
(27)

for some Borel non-atomic probability measure $\mu$ on $\mathbb{R}$ with
$\text{supp}(\mu)\subset A.$ More precisely, the supremum is taken over all
Borel non-atomic probability measures, supported on A and satisfying (27).

It is known (see, for example, [15, 16]) that for any Borel set
$A\subset\mathbb{R}$

$\dim_{F}A\leq\dim_{H}A.$ (28)

A set $A\subset\mathbb{R},$ for which $\dim_{F}A=\dim_{H}A,$ is called a Salem
set.

For the set of inhomogeneous $\psi$-well approximable real numbers
$W_{\mathcal{A}}(\gamma;\psi)$ an upper bound of the Hausdorff dimension is
known (see, for example, [9]):

$\dim_{H}W_{\mathcal{A}}(\gamma;\psi)\leq\min\left\\{\tau(\mathcal{A},\psi),1\right\\},$
(29)

where

$\tau(\mathcal{A},\psi)=\inf\left\\{\eta\geq
0\,\,:\,\,\sum_{q\in\mathcal{A}}q\left(\frac{\psi(q)}{q}\right)^{\eta}<\infty\right\\}.$
(30)

The next theorem gives a nontrivial result about the Fourier dimension of the
set of inhomogeneous $\psi$-well approximable real numbers lying inside some
other set $A$ and follows from Theorem 1.

######  Theorem 3.

Let $A\subset\mathbb{R}$ be a Borel set and let $\dim_{F}A=d\neq 0$. Let
$\alpha\in(0,1)$, and let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an
increasing $\alpha$-separated sequence of natural numbers. Assume that, for
some $\delta>0$ and $n_{0}\in\mathbb{N}$, the sequence $\mathcal{A}$ has the
growth rate

$q_{n}\geq n^{\frac{4}{d}+\delta},\quad n\geq n_{0}.$ (31)

Let $\psi:\mathbb{N}\rightarrow\mathbb{I}$ be an approximation function.
Assume that

$\Psi(N):=\sum_{n=1}^{N}\psi(q_{n})\to\infty\quad\text{as}\quad N\to\infty.$
(32)

Then, for any $\gamma\in\mathbb{I}$,

$\dim_{F}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)=d.$

###### Proof of Theorem 3.

Firstly note that the definition of the Fourier dimension implies that for any
$A_{1},A_{2}\subset\mathbb{R}$

$\dim_{F}\left(A_{1}\cap
A_{2}\right)\leq\min\left\\{\dim_{F}A_{1},\dim_{F}A_{2}\right\\}.$

Therefore,

$\dim_{F}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)\leq\dim_{F}A=d.$

So, in order to prove Theorem 3, we need to show that

$\dim_{F}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)\geq d,$ (33)

which means that the intersection $W_{\mathcal{A}}(\gamma;\psi)\cap A$
supports some Borel non-atomic probability measure with a suitable decay rate
of its Fourier transform.

Let us fix a positive number
$\widetilde{\varepsilon}<\frac{d^{2}\delta}{8+2d\delta}.$ Since
$\dim_{F}A=d\neq 0,$ then, for any $\varepsilon\in(0,\tilde{\varepsilon})$,
there exists a non-atomic probability measure $\mu_{A}^{(\varepsilon)},$
supported on $A$ and satisfying

$|\widehat{\mu_{A}^{(\varepsilon)}}(t)|=O\left(|t|^{-\frac{d}{2}+\varepsilon}\right),\quad\text{as}\quad|t|\to\infty.$
(34)

Put

$h(t)=t^{-\frac{d}{2}+\widetilde{\varepsilon}},\quad t\in\mathbb{N}.$

Now, for any $\varepsilon\in(0,\tilde{\varepsilon})$ we can apply Theorem 1,
choosing $\mu=\mu_{A}^{(\varepsilon)},$ $F=A$ and $h$ as above. Conditions (9)
and (10) of Theorem 1 follow from (34) and (31) respectively. By Theorem 1, we
get that, for any $\varepsilon\in(0,\tilde{\varepsilon})$ and for
$\mu_{A}^{(\varepsilon)}$-almost all $x\in A$, equality (11) holds true.
Taking into account (32), from (11) we get that

$R(x,N)\to\infty\;\text{ as
}N\to\infty,\quad\text{for}\,\,\mu_{A}^{(\varepsilon)}\text{-almost
all}\,\,x\in A,$

so for $\mu_{A}^{(\varepsilon)}$-almost all $x\in A$ we have $x\in
W_{\mathcal{A}}(\gamma;\psi).$ So, for any
$\varepsilon\in(0,\tilde{\varepsilon})$, we have

$\mu_{A}^{(\varepsilon)}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)=1.$

For any $\varepsilon\in(0,\tilde{\varepsilon})$ consider a measure space
$(\mathbb{R},\mathfrak{B}(\mathbb{R}),\mu_{A}^{(\varepsilon)}).$ Put
$B=W_{\mathcal{A}}(\gamma;\psi)\cap A.$ Now we modify measure
$\mu_{A}^{(\varepsilon)}$ introducing a new Borel measure

$\mu_{B}^{(\varepsilon)}(C):=\mu_{A}^{(\varepsilon)}(C\cap B),\quad\forall
C\in\mathfrak{B}(\mathbb{R}).$

Observing the full measure statement $\mu_{A}^{(\varepsilon)}(B)=1$ and the
fact that $\mu_{A}^{(\varepsilon)}$ is non-atomic and probability we deduce
that new measure space
$(\mathbb{R},\mathfrak{B}(\mathbb{R}),\mu_{B}^{(\varepsilon)})$ is a
probability space with a non-atomic measure $\mu_{B}^{(\varepsilon)}.$ From
the definition of measures $\mu_{B}^{(\varepsilon)}$ it follows that

$\text{supp}(\mu_{B}^{(\varepsilon)})\subset
B,\quad\forall\varepsilon\in(0,\tilde{\varepsilon}).$

Moreover, for any $t\in\mathbb{Z}$ and $\varepsilon\in(0,\tilde{\varepsilon})$

$\int_{\mathbb{R}}e^{-2\pi
itx}\,\mathrm{d}\mu_{A}^{(\varepsilon)}(x)=\int_{B}+\int_{\mathbb{R}\setminus
B}=\int_{B}e^{-2\pi itx}\,\mathrm{d}\mu_{A}^{(\varepsilon)}(x)=$
$=\int_{B}e^{-2\pi
itx}\,\mathrm{d}\mu_{B}^{(\varepsilon)}(x)=\int_{B}+\int_{\mathbb{R}\setminus
B}=\int_{\mathbb{R}}e^{-2\pi itx}\,\mathrm{d}\mu_{B}^{(\varepsilon)}(x),$

so $\mu_{B}^{(\varepsilon)}$ has the same decay rate of Fourier transform, as
$\mu_{A}^{(\varepsilon)}.$ Therefore, observing (34) with
$\mu_{B}^{(\varepsilon)}$ instead of $\mu_{A}^{(\varepsilon)},$

$\dim_{F}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)\geq d,$

so (33) holds. This finishes the proof of Theorem 3. ∎

The following two corollaries immediately follows from Theorem 3 and
inequalities

$\dim_{F}\left(A\cap W\right)\leq\dim_{H}\left(A\cap
W\right)\leq\min\left\\{\dim_{H}A,\dim_{H}W\right\\}\leq\dim_{H}A.$ (35)

In the above inequalities (35) sets $A$ and $W$ are two Borel subsets of
$\mathbb{R},$ these inequalities follow from (28) and definitions of Hausdorff
and Fourier dimensions.

######  Corollary 2.

If, under all conditions of Theorem 3, we also have that $A$ is a Salem set,
then the intersection $W_{\mathcal{A}}(\gamma;\psi)\cap A$ is also a Salem set
and

$\dim_{H}\left(W_{\mathcal{A}}(\gamma;\psi)\cap
A\right)=\dim_{F}\left(W_{\mathcal{A}}(\gamma;\psi)\cap A\right)=d.$

######  Remark 3.

In view of Corollary 2, for any given $d\in(0,1]$ we can build Salem subsets
$W$ of the set of inhomogeneous $\psi$-well approximable real numbers
$W_{\mathcal{A}}(\gamma;\psi)$ with $\dim_{H}W=d$. For example, in [9]
Hambrook proved that if the denominators sequence $\mathcal{A}_{1}$ satisfies

$\sum_{q\in\mathcal{A}_{1}}\frac{1}{q}=\infty,$

and for the approximation function $\psi_{1}$ we have

$\liminf_{M\to\infty}\frac{-\log\psi_{1}(M)}{\log
M}=\limsup_{M\to\infty}\frac{-\log\psi_{1}(M)}{\log M}=\lambda,$ (36)

then for any $\gamma_{1}\in\mathbb{I}$ the set
$W_{1}=W_{\mathcal{A}_{1}}(\gamma_{1};\psi_{1})$ is Salem with

$\dim_{H}W_{1}=\dim_{F}W_{1}=\min\\{\frac{2}{1+\lambda},1\\}.$ (37)

So, the subset $W=W_{\mathcal{A}}(\gamma;\psi)\cap
W_{\mathcal{A}_{1}}(\gamma_{1};\psi_{1}),$ whose elements are both $\psi$ and
$\psi_{1}$-well approximable (with different denominators sequences), is a
Salem set. Of course, this readily extends to finite intersections of the sets
of well-approximale numbers.

######  Corollary 3.

Let $\alpha\in(0,1)$, and let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an
increasing $\alpha$-separated sequence of natural numbers with a growth rate

$q_{n}\geq n^{4+\delta}$ (38)

for some $\delta>0$. Assume that approximation function $\psi$ verifies (32).
Then,

$\dim_{H}W_{\mathcal{A}}(\gamma;\psi)=1.$

###### Proof.

It is well-known that for every $d\in(0,1]$ there exists a Salem set with
Fourier dimension $d$. For example, one could use the set $W_{1}$ mentioned in
(37) with $\lambda=2/d-1$. Then, the result follows from Theorem 3 with
$A=W_{1}$ (and the use of inequalities (35)). ∎

######  Remark 4.

Corollary 3 provides an inhomogeneous Khintchine theorem for a wide range of
sequences of restraint denominators (more precisely, it provides the
divergence case of Khintchine’s theorem, but the convergence case always
readily follows from classical Borel-Cantelli lemma). Homogeneous version of
this result was previously established by Rynne [18]. He shows that

$\dim_{H}W_{\mathcal{A}}(0;\psi)=\min\left\\{\tau(\mathcal{A},\psi),1\right\\},$

where $\tau(\mathcal{A},\psi)$ is defined by (30). Note that (32) implies
$\tau(\mathcal{A},\psi)\geq 1,$ so in homogeneous case, under (32), we have
$\dim_{H}W_{\mathcal{A}}(0;\psi)=1.$ Inhomogeneous case is not so well
studied. The existing results cover $\mathcal{A}=\mathbb{N}$ and, more
generally, not too sparse subsets of $\mathbb{N}$ (verifying the condition
$\sum_{q\in\mathcal{A}_{1}}\frac{1}{q}=\infty$). Corollary 3, complimentarily,
gives inhomogeneous results for sufficiently sparse denominators sequence
(verifying (31)).

Of course, it would be very desirable to extend the results of Theorem 3
beyond the divergence condition (32). This could be done, at least partially,
by using a powerful Mass Transference Principle. Theorem 4 below states a
particular case that we use of a general result [2][Theorem 3].

######  Theorem 4.

Let $X\subset\mathbb{R}$ and let $\lambda,\mu>0$. Let
$\left(B_{i}\right)_{i\in\mathbb{N}}$ be a sequence of balls in $X$
(considered as a metric space with the evident distance $d(x,y)=|x-y|$) such
that radii of $B_{i}$ tend to 0 as $i\to\infty$. For each ball
$B_{i}=B_{i}(x_{i},r_{i})$ define
$B_{i}^{\lambda/\mu}:=B\left(x_{i},r_{i}^{\lambda/\mu}\right)$.

Assume that, for every ball $B\subset X$,

$\dim_{H}\left(B\cap\limsup_{i\to\infty}B_{i}^{\lambda/\mu}\right)\geq\mu.$

Then, for every ball $B\subset X$,

$\dim_{H}\left(B\cap\limsup_{i\to\infty}B_{i}\right)\geq\min\left(\lambda,\dim_{H}X\right).$

Theorem 5 below could be considered as a partial extension of Theorem 3. It
provides a non-trivial lower bound for Hausdorff dimension of intersection of
the set of well-approximable numbers with other sets.

######  Theorem 5.

Let $A\subset\mathbb{R}$ be a Borel set and let $\dim_{F}A=d\neq 0$. Let
$\alpha\in(0,1)$ and let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an
increasing $\alpha$-separated sequence of natural numbers. Assume that, for
some $\delta>0$ and $n_{0}\in\mathbb{N}$, the sequence $\mathcal{A}$ has the
growth rate (38). Let $\psi:\mathbb{N}\rightarrow\mathbb{I}$ be an
approximation function. Then, for any $\gamma\in\mathbb{I}$,

$\dim_{H}\left(W_{\mathcal{A}}(\gamma;\psi)\cap
A\right)\geq\tau(\mathcal{A},\psi)\cdot d,$

where $\tau(\mathcal{A},\psi)$ is defined by (30).

###### Proof.

The result follows from Theorem 3, inequalities (35) and Theorem 4 with
$\lambda=d\cdot\tau(\mathcal{A},\psi)$ and $\mu=d$. ∎

######  Example 3.

Let $\lambda\geq 0$, $\gamma_{1}\in\mathbb{I}$ and let $\psi_{1}$ and
$\mathcal{A}_{1}$ be the same as in Remark 3 (for the sake of even further
concretness, one could take $\psi_{1}:t\to t^{-\lambda}$ and
$\mathcal{A}_{1}=\mathbb{Z}$). As it is discussed in Remark 3, in this case
$W_{\mathcal{A}_{1}}(\gamma_{1},\psi_{1})$ is a Salem set verifying

$\dim_{H}W_{\mathcal{A}_{1}}(\gamma_{1},\psi_{1})=\dim_{F}W_{\mathcal{A}_{1}}(\gamma_{1},\psi_{1})=\frac{2}{1+\lambda}.$

Further, let $a\in\mathbb{N}$, $a\geq 2$ and let
$\mathcal{A}=\left(a^{n}\right)_{n\in\mathbb{N}}$. It is not hard to verify
that the sequence $\mathcal{A}$ is $1/2$-separated. Then, it follows from
Theorem 5 that, for any approximating function $\psi$,

$\dim_{H}\left(W_{\mathcal{A}_{1}}(\gamma_{1},\psi_{1})\cap
W_{\mathcal{A}}(\gamma,\psi)\right)\geq\frac{2\cdot\tau(\mathcal{A},\psi)}{1+\lambda}.$

The following corollary extends Corollary 3, providing thus a Järnik-type
theorem for a wide range on sequences of restraint denominators.

######  Corollary 4.

Let $\mathcal{A}=(q_{n})_{n\in\mathbb{N}}$ be an increasing $\alpha$-separated
sequence of natural numbers, for some $\alpha\in(0,1)$, and assume that, for
some $\delta>0$ and $n_{0}\in\mathbb{N}$, the sequence $\mathcal{A}$ has the
growth rate (38). Let $\psi:\mathbb{N}\rightarrow\mathbb{I}$ be an
approximation function. Then, for any $\gamma\in\mathbb{I}$,

$\dim_{H}\left(W_{\mathcal{A}}(\gamma;\psi)\right)=\tau(\mathcal{A},\psi),$

where $\tau(\mathcal{A},\psi)$ is defined by (30).

###### Proof.

Because of (29), we need to prove only the lower bound

$\dim_{H}W_{\mathcal{A}}(\gamma;\psi)\geq\tau(\mathcal{A},\psi).$

But this result follows from Theorem 5 with the remark that the interval
$A=[0,1]$ has Fourier dimension 1. ∎

##  5 Establishing the main result

The proof of Theorem 1 is based on the following lemma [10, Lemma 1.5].

######  Lemma 1.

Let $(X,\mathcal{B},\mu)$ be a probability space, let
$(f_{n}(x))_{n\in\mathbb{N}}$ be a sequence of non-negative $\mu$-measurable
functions defined on $X$, and $(f_{n})_{n\in\mathbb{N}},\
(\phi_{n})_{n\in\mathbb{N}}$ be sequences of real numbers such that

$0\leq f_{n}\leq\phi_{n}\hskip 19.91692pt(n=1,2,\ldots).$

Suppose that for arbitrary $a,b\in\mathbb{N}$ with $a<b$, we have

$\int_{X}\left(\sum_{n=a}^{b}\big{(}f_{n}(x)-f_{n}\big{)}\right)^{2}\mathrm{d}\mu(x)\,\leq\,C\\!\sum_{n=a}^{b}\phi_{n}$
(39)

for an absolute constant $C>0$. Then, for any given $\varepsilon>0$, we have

$\sum_{n=1}^{N}f_{n}(x)\,=\,\sum_{n=1}^{N}f_{n}\,+\,O\left(\Phi(N)^{1/2}\log^{\frac{3}{2}+\varepsilon}\Phi(N)+\max_{1\leq
k\leq N}f_{k}\right)$ (40)

for $\mu$-almost all $x\in X$, where $\Phi(N):=\sum\limits_{n=1}^{N}\phi_{n}$.

A mechanism of applying Lemma 1 for proving similar to our counting results
can be found in [17]. We will move in parallel to their proof taking into
account that, unlike [17], we do not have a growth conditions on the sequence
$\mathcal{A}=(q_{n})_{n\in\mathbb{N}}.$ and decay rate for $\hat{\mu}(t)$,
which are replaced in our paper by balance condition (10).

###  5.1 Choosing $f_{n}(x)$ and $f_{n}$ from Lemma 1

Let us consider Lemma 1 with

$X:=\mathbb{I}\,,\qquad
f_{n}(x):=\chi_{{}_{E(q_{n},\gamma,\psi)}}(x)\qquad{\rm and}\qquad
f_{n}=2\psi(q_{n})\,,$ (41)

It follows from (5) that, with this choice of $f_{n}(x)$,

$\text{the l.h.s. of }\eqref{ebc_conclusion}\ =\ R(x,N).$

Note that $f_{n}(x)^{2}=f_{n}(x),\quad x\in\mathbb{I},$, so, for any
$a,b\in\mathbb{N}$ with $a<b$,

$\displaystyle\left(\sum_{n=a}^{b}(f_{n}(x)-f_{n})\right)^{2}$
$\displaystyle=\
\left(\sum_{n=a}^{b}f_{n}(x)\right)^{2}\,+\,\left(\sum_{n=a}^{b}f_{n}\right)^{2}-\
2\sum_{n=a}^{b}f_{n}(x)\cdot\sum_{n=a}^{b}f_{n}$ $\displaystyle=\
\sum_{n=a}^{b}f_{n}(x)$ $\displaystyle+\ 2\mathop{\sum\sum}_{a\leq m<n\leq
b}f_{m}(x)f_{n}(x)\ +\ \left(\sum_{n=a}^{b}f_{n}\right)^{2}\ -\
2\sum_{n=a}^{b}f_{n}\cdot\sum_{n=a}^{b}f_{n}(x)\,,\vspace{2mm}$

and so it follows that

$\displaystyle\text{the l.h.s. of }\eqref{ebc_condition1}\ =\
\sum_{n=a}^{b}\mu(E_{n})\,+\,2$ $\displaystyle\mathop{\sum\sum}_{a\leq m<n\leq
b}\mu(E_{m}\cap E_{n})-$ (42) $\displaystyle-\
4\sum_{n=a}^{b}\psi(q_{n})\left(\sum_{n=a}^{b}\mu(E_{n})-\sum_{n=a}^{b}\psi(q_{n})\right),$

here we used the following short notation $E_{n}:=E(q_{n},\gamma,\psi),\quad
n\in\mathbb{N}.$

Relation (42) shows that we need to obtain a ‘good’ estimates of the measure
of sets $E_{n}=E(q_{n},\gamma,\psi)$ and the measure of their intersections to
satisfy the condition (39).

###  5.2 Estimating the measure of the sets $E_{n}$ and their intersections

In this section we present different estimates that help us to prove the main
result. We will use the Fourier analysis estimating the measure of sets
$E_{n}$ and their intersections.

Let $\varepsilon$ and $\delta$ be real numbers such that $0<\varepsilon\leq 1$
and $0<\delta<1/4$. Let $\chi_{\delta}:\mathbb{I}\to\mathbb{R}$ be the
characteristic function defined by

$\chi_{\delta}(x):=\begin{cases}1\ \text{ if }\
\|x\|\leq\delta,\\\\[4.30554pt] 0\ \text{ if }\ \|x\|>\delta\,,\end{cases}$

and let $\chi_{\delta,\varepsilon}^{+}:\mathbb{I}\to\mathbb{R}$ and
$\chi_{\delta,\varepsilon}^{-}:\mathbb{I}\to\mathbb{R}$ be the continuous
upper and lower approximations of $\chi_{\delta}$ given by

$\chi_{\delta,\varepsilon}^{+}(x):=\begin{cases}1&\text{\ if }\
\|x\|\leq\delta,\\\\[4.30554pt]
1+\dfrac{1}{\delta\varepsilon}(\delta-\|x\|)&\text{\ if }\
\delta<\|x\|\leq(1+\varepsilon)\delta\\\\[5.59721pt] 0&\text{\ if }\
\|x\|>(1+\varepsilon)\delta\,,\end{cases}$

and

$\chi_{\delta,\varepsilon}^{-}(x):=\begin{cases}1&\text{\ if }\
\|x\|\leq(1-\varepsilon)\delta\\\\[4.30554pt]
\dfrac{1}{\delta\varepsilon}(\delta-\|x\|)&\text{\ if }\
(1-\varepsilon)\delta<\|x\|\leq\delta\\\\[5.59721pt] 0&\text{\ if }\
\|x\|>\delta\,.\end{cases}$

Clearly, both $\chi_{\delta,\varepsilon}^{+}$ and
$\chi_{\delta,\varepsilon}^{-}$ are periodic functions with period 1. Next,
given a real positive function $\psi:\mathbb{N}\rightarrow\mathbb{I}$ and any
integer $q\geq 4$, consider the functions $W_{q,\gamma,\varepsilon,\psi}^{+}$
and $W_{q,\gamma,\varepsilon,\psi}^{-}$ defined by

$W_{q,\gamma,\varepsilon}^{+}(x)=W_{q,\gamma,\varepsilon,\psi}^{+}(x)\ :=\
\Big{(}\sum_{p=0}^{q-1}\delta_{\frac{p+\gamma}{q}}(x)\Big{)}*\chi_{\frac{\psi(q)}{q},\varepsilon}^{+}(x)$
(43)

and

$W_{q,\gamma,\varepsilon}^{-}(x)=W_{q,\gamma,\varepsilon,\psi}^{-}(x)\ :=\
\Big{(}\sum_{p=0}^{q-1}\delta_{\frac{p+\gamma}{q}}(x)\Big{)}*\chi_{\frac{\psi(q)}{q},\varepsilon}^{-}(x)\,,\vspace{2mm}$

where $*$ denotes convolution and $\delta_{x}$ denotes the Dirac delta-
function at the point $x\in\mathbb{R}$.

It occurs that

$\displaystyle
W_{q,\gamma,\varepsilon}^{+}(x)=\sum_{p=0}^{q-1}\chi_{\frac{\psi(q)}{q},\varepsilon}^{+}\left(\textstyle{x-\frac{p+\gamma}{q}}\right)$

and

$\displaystyle
W_{q,\gamma,\varepsilon}^{-}(x)=\sum_{p=0}^{q-1}\chi_{\frac{\psi(q)}{q},\varepsilon}^{-}\left(\textstyle{x-\frac{p+\gamma}{q}}\right)\,.$

It thus follows that for any $0<\varepsilon\leq 1$ and any integer $q\geq 4$,

$\int_{0}^{1}W_{q,\gamma,\varepsilon}^{-}(x)\mathrm{d}\mu(x)\;\leq\;\mu(E(q,\gamma,\psi))\;\leq\;\int_{0}^{1}W_{q,\gamma,\varepsilon}^{+}(x)\mathrm{d}\mu(x).\vspace{2mm}$
(44)

Now we need to consider the Fourier series expansions

$\sum_{k\in\mathbb{Z}}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k)\exp(2\pi
kix)$

of $W_{q,\gamma,\varepsilon}^{+}$ and $W_{q,\gamma,\varepsilon}^{-}$.

The values and basic estimates of the Fourier coefficients
$\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k)$ are presented in [17] (see Lemma
1 in [17]). For the convenience of the reader, we collect them in the
following proposition.

######  Proposition 2.

Let $0<\varepsilon,\tilde{\varepsilon}\leq 1$ and
$\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k),\quad k\in\mathbb{Z},$ are the
Fourier coefficients of $W_{q,\gamma,\varepsilon}^{+}$ and
$W_{q,\gamma,\varepsilon}^{-}$. Then, for any integers $q,r\geq 4:$

_(i)_ for $k\neq 0$,

$\widehat{W}_{q,\gamma,\varepsilon}^{+}(k)\,=\,\begin{cases}\exp\left(-\dfrac{2\pi
ik\gamma}{q}\right)\dfrac{q\left(\cos(2\pi k\psi(q)q^{-1})-\cos(2\pi
k\psi(q)q^{-1}(1+\varepsilon))\right)}{2\pi^{2}k^{2}\psi(q)q^{-1}\varepsilon}&\text{\
if }\ q\mid k\\\\[12.91663pt] 0&\text{\ if }\ q\nmid k\,,\end{cases}\,$ (45)

and for $k=0$,

$\widehat{W}_{q,\gamma,\varepsilon}^{+}(0)=(2+\varepsilon)\,\psi(q)\,;\vspace{2mm}$
(46)

_(ii)_ for $k\neq 0$,

$\widehat{W}_{q,\gamma,\varepsilon}^{-}(k)\,=\,\begin{cases}\exp\left(-\dfrac{2\pi
ik\gamma}{q}\right)\dfrac{q\left(\cos(2\pi
k\psi(q)q^{-1}(1-\varepsilon))-\cos(2\pi
k\psi(q)q^{-1})\right)}{2\pi^{2}k^{2}\psi(q)q^{-1}\varepsilon}&\text{\ if }\
q\mid k\\\\[12.91663pt] 0&\text{\ if }\ q\nmid k\,,\vspace{2mm}\end{cases}$
(47)

and for $k=0$,

$\widehat{W}_{q,\gamma,\varepsilon}^{-}(0)=(2-\varepsilon)\,\psi(q)\,;\vspace{2mm}$
(48)

_(iii)_ for any $s\in\mathbb{Z}\setminus\\{0\\}$

$\displaystyle\left|\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(sq)\right|$
$\displaystyle\leq$ $\displaystyle(2+\varepsilon)\,\psi(q),$ (49)
$\displaystyle\left|\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(sq)\right|$
$\displaystyle\leq$ $\displaystyle\frac{1}{\pi^{2}s^{2}\psi(q)\varepsilon}.$
(50)

_(iv)_

$\displaystyle\sum_{s\in\mathbb{Z}}\big{|}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(sq)\big{|}\
<\ \frac{3}{\varepsilon^{1/2}}$ (51)

and

$\sum_{s\in\mathbb{Z}}\,\sum_{t\in\mathbb{Z}}\,\big{|}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(sq)\big{|}\
\big{|}\widehat{W}_{r,\gamma,\tilde{\varepsilon}}^{\pm}(tr)\big{|}\ \leq\
\frac{9}{\varepsilon^{1/2}\cdot\tilde{\varepsilon}^{1/2}}\,.$ (52)

By (51),
$\sum\limits_{k\in\mathbb{Z}}\big{|}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k)\big{|}<\infty$,
so the Fourier series

$\sum_{k\in\mathbb{Z}}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k)\exp(2\pi
kix)$

converges uniformly to $W_{q,\gamma,\varepsilon}^{\pm}(x)$ for all
$x\in\mathbb{I}$. Hence, it follows that

$\displaystyle\int_{0}^{1}W_{q,\gamma,\varepsilon}^{\pm}(x)\;\mathrm{d}\mu(x)\;=\;\sum_{k\in\mathbb{Z}}\,\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(k)\;\widehat{\mu}(-k)\,.$

This together with (44), (46), (48) and the fact that $\widehat{\mu}(0)=1$,
implies that

$\begin{array}[]{ll}\mu(E(q,\gamma,\psi))\ \leq\ (2+\varepsilon)\,\psi(q)\
+\displaystyle{\sum_{k\in\mathbb{Z}\setminus\\{0\\}}}\widehat{W}_{q,\gamma,\varepsilon}^{+}(k)\;\widehat{\mu}(-k)\\\\[25.83325pt]
\mu(E(q,\gamma,\psi))\ \geq\ (2-\varepsilon)\,\psi(q)\
+\displaystyle{\sum_{k\in\mathbb{Z}\setminus\\{0\\}}}\widehat{W}_{q,\gamma,\varepsilon}^{-}(k)\;\widehat{\mu}(-k)\,.\end{array}\,$
(53)

####  5.2.1 Estimating the sums from measures of $E_{n}$

Now we are ready to prove the following estimates for the sums of
$\mu(E_{n}).$

######  Lemma 2.

Under the conditions of Theorem 1, we have, for arbitrary $a,b\in\mathbb{N}$
with $a<b$,

$\sum_{n=a}^{b}\mu(E(q_{n},\gamma,\psi))=2\sum_{n=a}^{b}\psi(q_{n})\,+\,O\left(\min\Big{(}1,\sum_{n=a}^{b}\psi(q_{n})\Big{)}\right).$
(54)

We need the following well known statement in order to prove Lemma 2. We’ll
leave this statement without citation because, in our opinion, it belongs to
the mathematical folklore.

######  Lemma 3.

Let $(a_{n},n\in\mathbb{N})$ be a sequence of nonnegative real numbers
verifying $a_{1}>0$ and

$S_{n}=a_{1}+a_{2}+\ldots+a_{n},\quad n\in\mathbb{N}.$

Then for any real $\xi>0$ the series

$\sum_{n=1}^{\infty}\frac{a_{n}}{S_{n}^{1+\xi}}$

converges.

###### Proof of Lemma 3.

Firstly, note that, for any $n\in\mathbb{N}$,

$\int_{S_{n-1}}^{S_{n}}\frac{dx}{x^{1+\xi}}\geq\int_{S_{n-1}}^{S_{n}}\frac{dx}{S_{n}^{1+\xi}}=\frac{a_{n}}{S_{n}^{1+\xi}}.$
(55)

To prove the convergency of the series

$\sum_{n=1}^{\infty}\frac{a_{n}}{S_{n}^{1+\xi}}$

consider it partial sum

$B_{N}=\sum_{n=1}^{N}\frac{a_{n}}{S_{n}^{1+\xi}}.$

In view of (55), for any $N\in\mathbb{N},$

$B_{N}=\frac{a_{1}}{S_{1}^{1+\xi}}+\frac{a_{2}}{S_{2}^{1+\xi}}+\frac{a_{3}}{S_{3}^{1+\xi}}+\ldots+\frac{a_{N}}{S_{N}^{1+\xi}}\leq$
$\leq\frac{a_{1}}{S_{1}^{1+\xi}}+\int_{S_{1}}^{S_{2}}\frac{dx}{x^{1+\xi}}+\int_{S_{2}}^{S_{3}}\frac{dx}{x^{1+\xi}}+\ldots+\int_{S_{N-1}}^{S_{N}}\frac{dx}{x^{1+\xi}}=\frac{1}{a_{1}^{\xi}}+\int_{S_{1}}^{S_{N}}\frac{dx}{x^{1+\xi}}=$
$=\frac{1}{a_{1}^{\xi}}+\frac{1}{\xi}\left(\frac{1}{S_{1}^{\xi}}-\frac{1}{S_{N}^{\xi}}\right)\leq\frac{1}{a_{1}^{\xi}}\left(1+\frac{1}{\xi}\right).$

So, the sequence of partial sums $(B_{N},N\in\mathbb{N})$ for the series with
nonnegative elements is bounded, which implies the convergency of this series.
∎

Now we can prove Lemma 2.

###### Proof of Lemma 2.

We prove Lemma 2 under a weaker condition on number $\rho$ from (9), compared
to Theorem 1, namely $\rho>\frac{3}{2}$. For any given sequence of real
numbers $(\varepsilon_{n})_{n=a}^{b}$ in $(0,1]$, it follows from (53) that

$\left|\mu(E_{n})-2\psi(q_{n})\right|\leq\psi(q_{n})\varepsilon_{n}+\max_{\circ\in\\{+,-\\}}\left|\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{\circ}(k)\widehat{\mu}(-k)\right|,\quad
a\leq n\leq b.$ (56)

It follows from (45) and (47) that
$\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{\pm}(k)=0$ unless $k=sq_{n}$ for
some integer $s.$ Also from (10) we have, that $|\widehat{\mu}(sq_{n})|\ll
n^{-\rho},$ where $\rho>\frac{3}{2}.$ So, using (51),

$\left|\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{\pm}(k)\widehat{\mu}(-k)\right|\leq\sum_{s\in\mathbb{Z}\setminus\\{0\\}}\big{|}\widehat{W}_{q,\gamma,\varepsilon}^{\pm}(sq)\big{|}\big{|}\widehat{\mu}(sq_{n})\big{|}\ll\frac{3}{n^{\rho}\varepsilon_{n}^{\frac{1}{2}}}.$
(57)

Now, in view of (56) and (57),

$\left|\sum_{n=a}^{b}\mu(E_{n})-2\sum_{n=a}^{b}\psi(q_{n})\right|\ll\sum_{n=a}^{b}\psi(q_{n})\varepsilon_{n}+\sum_{n=a}^{b}\frac{3}{n^{\rho}\varepsilon_{n}^{\frac{1}{2}}},$
(58)

therefore, for $\varepsilon_{n}=1,$ $a\leq n\leq b$,

$\left|\sum_{n=a}^{b}\mu(E_{n})-2\sum_{n=a}^{b}\psi(q_{n})\right|\ll\sum_{n=a}^{b}\psi(q_{n})\,.$
(59)

To complete the proof we need to get one more upper bound, namely

$\left|\sum_{n=a}^{b}\mu(E_{n})-2\sum_{n=a}^{b}\psi(q_{n})\right|\ll 1.$ (60)

We can deduce the upper bound (60) again from (58) choosing suitable values of
$(\varepsilon_{n})_{n=a}^{b}.$ Since $\rho>\frac{3}{2},$ let’s choose a real
number $\xi\in(0,\,2\rho-3).$ For any $n\in\mathbb{N}$ let

$\Psi(n):=\sum_{k=1}^{n}\psi(q_{k})\quad{\rm
and}\quad\varepsilon_{n}:=\min\left(1,\Psi(n)^{-1-\xi}\right).$

By definition, $|\psi(q_{n})|\leq 1$ and so $\varepsilon_{n}^{-1}\leq
n^{1+\xi}\,.$ So, for the second term on the r.h.s. of (58) we have

$\sum_{n=a}^{b}\frac{3}{n^{\rho}\varepsilon_{n}^{1/2}}\,\leq\,\sum_{n=1}^{\infty}\frac{3}{n^{\rho-\frac{\xi}{2}-\frac{1}{2}}}\,<\,\infty\,,$
(61)

since $\rho-\frac{\xi}{2}-\frac{1}{2}>1$ because of the choice of $\xi$.

It follows from Lemma 3 that, for $\xi>0$, the series

$\sum_{n=1}^{\infty}\frac{\psi(q_{n})}{\Psi(n)^{1+\xi}}$

converges, so

$\sum_{n=1}^{\infty}\frac{\psi(q_{n})}{\max\left(1,\Psi(n)^{1+\xi}\right)}$

converges also. Therefore,

$\sum_{n=a}^{b}\psi(q_{n})\varepsilon_{n}\,<\,\sum_{n=1}^{\infty}\frac{\psi(q_{n})}{\max\left(1,\Psi(n)^{1+\xi}\right)}\,<\,\infty\,.$
(62)

The upper bound (60) now follows from the inequalities (58), (61) and (62). ∎

####  5.2.2 Estimating the sums from measures of intersections $E_{n}\cap
E_{m}$

Estimating the sums from measures of intersections of the sets $E_{n}$ is
technically a more complicated work. Here we use the condition on
$\alpha$-separability (see Definition 2), which we don’t use proving Lemma 2.

Before formulating the statement on the upper bound for the sums from measures
of intersections we prove the following result, related to the case $\xi=0$ of
Lemma 3.

######  Lemma 4.

Let $(a_{n},n\in\mathbb{N})$ be a sequence of nonnegative real numbers,
$a_{1}\neq 0$ and

$S_{n}=a_{1}+a_{2}+\ldots+a_{n},\quad n\in\mathbb{N}.$

Then for any $N\in\mathbb{N}$ for the partial sums $B_{N}$ of the series

$\sum_{n=1}^{\infty}\frac{a_{n}}{S_{n}}$

we have

$B_{N}=\sum_{n=1}^{N}\frac{a_{n}}{S_{n}}\leq 1+\log(S_{N})-\log(a_{1}).$

###### Proof of Lemma 4.

Firstly note that for any $n\in\mathbb{N}$

$\int_{S_{n-1}}^{S_{n}}\frac{dx}{x}\geq\int_{S_{n-1}}^{S_{n}}\frac{dx}{S_{n}}=\frac{a_{n}}{S_{n}}.$
(63)

In view of (63), for any $N\in\mathbb{N},$

$B_{N}=\frac{a_{1}}{S_{1}}+\frac{a_{2}}{S_{2}}+\frac{a_{3}}{S_{3}}+\ldots+\frac{a_{N}}{S_{N}}\leq$
$\leq\frac{a_{1}}{S_{1}}+\int_{S_{1}}^{S_{2}}\frac{dx}{x}+\int_{S_{2}}^{S_{3}}\frac{dx}{x}+\ldots+\int_{S_{N-1}}^{S_{N}}\frac{dx}{x}=1+\int_{S_{1}}^{S_{N}}\frac{dx}{x}=$
$=1+\log(S_{N})-\log(S_{1})=1+\log(S_{N})-\log(a_{1}).$

∎

Now we can formulate and prove the statement on the upper bound for the sums
from measures of intersections.

######  Lemma 5.

Let for any $\tau>1$

$\psi(q_{n})\geq n^{-\tau},\quad n\in\mathbb{N}.$ (64)

Then, under all conditions of Theorem 1, for arbitrary $a,b\in\mathbb{N}$ with
$a<b$,

$2\mathop{\sum\sum}_{a\leq m<n\leq b}\mu(E_{m}\cap E_{n})\ \leq\
\left(\sum_{n=a}^{b}\mu(E_{n})\right)^{2}+O\left(\mathop{\sum\sum}_{a\leq
m<n\leq
b}(q_{m},q_{n})\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\right)\\\\[8.61108pt]
+O\left(\left(\sum_{n=a}^{b}\psi(q_{n})\right)\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)+\sum_{n=a}^{b}\psi(q_{n})\right),$
(65)

where $\log^{+}(x):=\max(0,\log(x)),\quad x>0.$

###### Proof of Lemma 5.

In order to receive suitable estimates of the measures of intersections we
starts with some Fourier analysis (once again, we move here in parallel with
the corresponding reasonings in [17]). So, let’s fix a sequence of real
numbers $(\varepsilon_{n})_{n\in\mathbb{N}}\subset(0,1]$ and consider
functions

$W_{m,n}^{+}(x):=W_{q_{m},\gamma,\varepsilon_{m}}^{+}(x)\cdot
W_{q_{n},\gamma,\varepsilon_{n}}^{+}(x),\quad
x\in\mathbb{R},\,m,n\in\mathbb{N},\,m<n,$ (66)

(recall that function $W_{q,\gamma,\varepsilon}^{+}$ is defined by (43)).
Then,

$\displaystyle\mu(E_{m}\cap E_{n})$ $\displaystyle\leq$
$\displaystyle\int_{0}^{1}W_{q_{m},\gamma,\varepsilon_{m}}^{+}(x)W_{q_{n},\gamma,\varepsilon_{n}}^{+}(x)\,\mathrm{d}\mu(x)$
$\displaystyle=$ $\displaystyle\int_{0}^{1}W_{m,n}^{+}(x)\mathrm{d}\mu(x).$

The Fourier coefficients of the product $W_{m,n}^{+}$ are convolutions

$\displaystyle\widehat{W}_{m,n}^{+}(k)$ $\displaystyle:=$
$\displaystyle\,\int_{0}^{1}W_{q_{m},\gamma,\varepsilon_{m}}^{+}(x)W_{q_{n},\gamma,\varepsilon_{n}}^{+}(x)\,\exp(-2\pi
kix)\mathrm{d}x$ (67) $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z}}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(j)\
\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(k-j),\quad k\in\mathbb{Z}.$

Moreover,
$\sum\limits_{k\in\mathbb{Z}}\big{|}\widehat{W}_{m,n}^{+}(k)\big{|}<\infty$,
so the Fourier series

$\sum_{k\in\mathbb{Z}}\widehat{W}_{m,n}^{+}(k)\exp(2\pi kix)$

converges uniformly to $W_{m,n}^{+}(x)$ for all $x\in\mathbb{I}$. Hence, it
follows that

$\displaystyle\int_{0}^{1}W_{m,n}^{+}(x)\;\mathrm{d}\mu(x)\;=\;\sum_{k\in\mathbb{Z}}\,\widehat{W}_{m,n}^{+}(k)\;\widehat{\mu}(-k)\,.$

So,

$\displaystyle\mu(E_{m}\cap E_{n})=\mu(E(q_{m},\gamma,\psi)\cap
E(q_{n},\gamma,\psi))$ $\displaystyle\leq$ (68)
$\displaystyle\leq\sum_{k\in\mathbb{Z}}\widehat{W}_{m,n}^{+}(k)\widehat{\mu}(-k)$
$\displaystyle=$
$\displaystyle\widehat{W}_{m,n}^{+}(0)+\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{m,n}^{+}(k)\widehat{\mu}(-k).\vspace{2mm}$

We consider the two terms on the right hand side of (68) separately. From (67)
with $k=0,$ (and refer to (3)) we have

$\displaystyle\widehat{W}_{m,n}^{+}(0)$
$\displaystyle:=\,\int_{0}^{1}W_{q_{m},\gamma,\varepsilon_{m}}^{+}(x)W_{q_{n},\gamma,\varepsilon_{n}}^{+}(x)\mathrm{d}x\leq$
(69)
$\displaystyle\leq\,\Big{|}E\left(q_{m},\gamma,(1+\varepsilon_{m})\cdot\psi\right)\cap
E\left(q_{m},\gamma,(1+\varepsilon_{n})\cdot\psi\right)\Big{|}\,,\vspace{2mm}$

where $|\,.\,|$ is Lebesgue measure. It is relatively straightforward to
verify (see [10, Equation 3.2.5]222Equation 3.2.5 in [10] as stated is not
correct – the ‘big O’ error term is missing. for the details) that for any
$q,q^{\prime}\in\mathbb{N}$

$|E(q,\gamma,\psi)\cap
E(q^{\prime},\gamma,\psi)|\,=\,4\psi(q)\psi(q^{\prime})+O\left((q,q^{\prime})\
\min\Big{(}\frac{\psi(q)}{q},\frac{\psi(q^{\prime})}{q^{\prime}}\Big{)}\right)\,.$

Hence, it follows that

$\Big{|}E\left(q_{m},\gamma,(1+\varepsilon_{m})\cdot\psi\right)\cap
E\left(q_{m},\gamma,(1+\varepsilon_{n})\cdot\psi\right)\Big{|}\,=\,4(1+\varepsilon_{m})(1+\varepsilon_{n})\psi(q_{m})\psi(q_{n})\\\\[4.30554pt]
+\ \
O\left((q_{m},q_{n})\min\Big{(}\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\Big{)}\right)\,.$

This together with (69) implies that

$\widehat{W}_{m,n}^{+}(0)\,\leq\,4(1+\varepsilon_{m})(1+\varepsilon_{n})\psi(q_{m})\psi(q_{n})+O\left((q_{m},q_{n})\min\Big{(}\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\Big{)}\right)\,,$
(70)

which give us the estimate of the first term on the right hand side of
(5.2.2).

We proceed with considering the second term, which we will denote by
$S_{m,n}$. In view of (45) and (67), it follows that

$\displaystyle S_{m,n}$ $\displaystyle:=$
$\displaystyle\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{m,n}^{+}(k)\widehat{\mu}(-k)=$
(71) $\displaystyle=$
$\displaystyle\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}}}_{sq_{m}-tq_{n}\neq
0}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(-(sq_{m}+tq_{n})\right).$

We decompose $S_{m,n}$ into three sums:

$S_{m,n}=S_{1}(m,n)+S_{2}(m,n)+S_{3}(m,n),\vspace{2mm}$

where

$\displaystyle S_{1}(m,n)$ $\displaystyle:=$
$\displaystyle\sum_{t\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(0)\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}(-tq_{n}),$
$\displaystyle S_{2}(m,n)$ $\displaystyle:=$
$\displaystyle\sum_{s\in\mathbb{Z}\setminus\\{0\\}}\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(0)\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(sq_{m})\widehat{\mu}(-sq_{m}),$
$\displaystyle S_{3}(m,n)$ $\displaystyle:=$
$\displaystyle\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}}_{sq_{m}+tq_{n}\neq
0}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(-(sq_{m}+tq_{n})\right).\vspace{2mm}$

Inequalities (46), (51) and balance condition (10) imply that, for any
$m,n\in\mathbb{N},$ $m<n$,

$|S_{1}(m,n)|\ \ll\
\frac{(2+\varepsilon_{m})\psi(q_{m})}{n^{\rho}}\sum_{t\in\mathbb{Z}}\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\
\ll\ \frac{\psi(q_{m})}{n^{\rho}\varepsilon_{n}^{1/2}}\,.$ (72)

Symmetrically, for any $m,n\in\mathbb{N},$ $m<n,$

$|S_{2}(m,n)|\ \ll\ \frac{\psi(q_{n})}{m^{\rho}\varepsilon_{m}^{1/2}}\,.$ (73)

Now we decompose $S_{3}$ into two sums:

$S_{3}(m,n)\,=\,S_{4}(m,n)+S_{5}(m,n),$

where

$S_{4}(m,n)\
:=\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}}_{|sq_{m}-tq_{n}|\geq
q_{m}^{\alpha}}\\!\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(sq_{m}-tq_{n}\right)$

and

$S_{5}(m,n)\
:=\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}}_{1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha}}\\!\\!\\!\\!\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(sq_{m}-tq_{n}\right),$
(74)

here $\alpha\in(0,1)$ is the constant from $\alpha$-separability condition of
Theorem 1. Regarding $S_{4}$, by making use of balance condition (10) with the
restriction $|sq_{m}-tq_{n}|\geq q_{m}^{\alpha}$ and inequality (52), it
follows that

$|S_{4}(m,n)|\ \ll\
\frac{1}{m^{\rho}}\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}\left|\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\right|\left|\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\right|\
\ll\ \frac{1}{m^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}\,.$ (75)

Finally, we decompose $S_{5}$ into two sums:

$S_{5}(m,n)\,=\,S_{6}(m,n)+E(m,n),$

where

$S_{6}(m,n):=\mathop{\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}}_{s>m^{3}/\psi(q_{m})}}_{1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha}}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(sq_{m}-tq_{n}\right)\,\vspace{2mm}$

and

$E(m,n):=\mathop{\mathop{\mathop{\sum\sum}_{s,t\in\mathbb{Z}\setminus\\{0\\}}}_{1\leq
s\leq
m^{3}/\psi(q_{m})}}_{1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha}}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(sq_{m}-tq_{n}\right).$

The restriction $1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha}$ implies that

$0<\left|s-t\frac{q_{n}}{q_{m}}\right|<1\,.$ (76)

Hence, if nonzero $s$ and $t$ satisfy (76) then both necessarily must have the
same sign and also for each fixed integer $s$ there exists a set $T_{s}$ of at
most two non-zero integers $t$ satisfying the restriction
$1\leq|sq_{m}-tq_{n}|<q_{m}^{\alpha}.$ So, we can write $S_{6}$ as a single
sum

$S_{6}(m,n)=\mathop{\mathop{\sum}_{s>m^{3}/\psi(q_{m}):}}_{t\in
T_{s}}\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(-sq_{m})\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})\widehat{\mu}\left(sq_{m}-tq_{n}\right).$

So, using the trivial bound $|\widehat{\mu}(t)|\leq 1$ together with (49) to
bound $|\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})|$ and (50) to
bound $|\widehat{W}_{q_{m},\gamma,\varepsilon_{n}}^{+}(sq_{m})|$, we obtain
that for any integers $1\leq m<n$

$\displaystyle|S_{6}(m,n)|$ $\displaystyle\ll$
$\displaystyle\mathop{\mathop{\sum}_{s>m^{3}/\psi(q_{m}):}}_{t\in
T_{s}}|\widehat{W}_{q_{m},\gamma,\varepsilon_{m}}^{+}(sq_{m})|\
|\widehat{W}_{q_{n},\gamma,\varepsilon_{n}}^{+}(tq_{n})|$ (77)
$\displaystyle\ll$
$\displaystyle\sum_{s>m^{3}/\psi(q_{m})}\frac{1}{s^{2}\psi(q_{m})\,\varepsilon_{m}}\psi(q_{n})\
\ll\ \frac{\psi(q_{n})}{m^{3}\,\varepsilon_{m}}\,.$

Working with $E(m,n)$ note, that in view of (64) the condition $s\leq
m^{3}/\psi(q_{m})$ implies that $s\leq m^{5}.$ This, together with the fact
that $(q_{n})_{n\in\mathbb{N}}$ is $\alpha$-separated implies that $E(m,n)$ is
empty sum. Thus,

$E(m,n)=0.$ (78)

So, from upper bounds (72), (73), (75), (77) and equality (78), for any fixed
$a,b\in\mathbb{N}$ with $a<b,$ and all natural numbers $m,n,$ with $a\leq
m<n\leq b:$

$|S_{m,n}|\ll\frac{\psi(q_{m})}{n^{\rho}\varepsilon_{n}^{1/2}}\,+\frac{\psi(q_{n})}{m^{\rho}\varepsilon_{m}^{1/2}}\,+\frac{1}{m^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}\,+\frac{\psi(q_{n})}{m^{3}\,\varepsilon_{m}}\,.$
(79)

Now, given $a\in\mathbb{N}$, define, for all $n\in\mathbb{N}$, $n\geq a$,

$\varepsilon_{n}:=\min\left(1,\,\left(\sum_{k=a}^{n}\psi(q_{k})\right)^{-1}\right).$
(80)

Then, since $\psi(k)\leq 1$ for all $k\in\mathbb{N}$,

$\varepsilon_{n}^{-1}\,\leq\,\max\left(1,n\right)\,\leq\,n,\quad n\geq a.$
(81)

Therefore,

$\sum_{n=a}^{\infty}\frac{1}{n^{\rho}\varepsilon_{n}^{1/2}}\,\leq\,\sum_{n=a}^{\infty}\frac{1}{n^{\rho-\frac{1}{2}}}\,<\,\infty\,,$

(recall that $\rho>2$), and so, it follows from (79) that for any
$a,b\in\mathbb{N},$ $a<b,$

$\mathop{\sum\sum}_{a\leq m<n\leq
b}|S_{m,n}|\,\ll\,\sum_{n=a}^{b}\psi(q_{n})+\mathop{\sum\sum}_{a\leq m<n\leq
b}\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}+\mathop{\sum\sum}_{a\leq
m<n\leq b}\frac{\psi(q_{n})}{m^{3}\,\varepsilon_{m}}\,.$ (82)

For the third sum in the right hand side of (82), from inequalities (81), we
have

$\mathop{\sum\sum}_{a\leq m<n\leq
b}\frac{\psi(q_{n})}{m^{3}\,\varepsilon_{m}}\leq\mathop{\sum\sum}_{a\leq
m<n\leq b}\frac{\psi(q_{n})}{m^{2}}\ll\sum_{n=a}^{b}\psi(q_{n}),\quad
a,b\in\mathbb{N},\,a<b.$ (83)

In order to estimate the second sum in the right hand side of (82) let’s
consider two cases.

Case 1: $\sum\limits_{k=a}^{b}\psi(q_{k})>1$. Then, by (80),

$\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}\leq\frac{1}{n^{\rho}}\sum_{k=a}^{b}\psi(q_{k}),$

and so, since $\rho>2,$

$\mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}\leq\left(\mathop{\sum\sum}\limits_{a\leq
m<n\leq
b}\frac{1}{n^{\rho}}\right)\cdot\left(\sum_{k=a}^{b}\psi(q_{k})\right)\ll\sum_{k=a}^{b}\psi(q_{n}).$
(84)

Case 2: $\sum\limits_{k=a}^{b}\psi(q_{k})\leq 1$. It follows, by (80), that
$\varepsilon_{n}=1$ for all $a\leq n\leq b$, therefore

$\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}=\frac{1}{n^{\rho}}.$

By using Lemma 5 with $\tau=\frac{\rho}{2}$, we get

$\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}=\frac{1}{n^{\rho}}\leq\frac{\psi(q_{n})}{n^{\frac{\rho}{2}}},\quad\
n\in\mathbb{N},$

and so

$\mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\frac{1}{n^{\rho}\varepsilon_{m}^{1/2}\varepsilon_{n}^{1/2}}\leq\mathop{\sum\sum}\limits_{a\leq
m<n\leq
b}\frac{\psi(q_{n})}{n^{\frac{\rho}{2}}}\leq\mathop{\sum\sum}\limits_{a\leq
m<n\leq b}\frac{\psi(q_{n})}{m^{\frac{\rho}{2}}}\ll\sum_{n=a}^{b}\psi(q_{n}).$
(85)

Both cases give us the same upper bound. So, estimates (83), (84) and (85),
together with (82), imply that, for all $a,b\in\mathbb{N}$ with $a<b$,

$\mathop{\sum\sum}_{a\leq m<n\leq
b}|S_{m,n}|\,\ll\,\sum_{n=a}^{b}\psi(q_{n})\,.$

Therefore, by (68),

$\mathop{\sum\sum}\limits_{a\leq m<n\leq b}\mu(E_{m}\cap
E_{n})\leq\mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\widehat{W}_{m,n}^{+}(0)+O\left(\sum_{n=a}^{b}\psi(q_{n})\right).\vspace{3ex}$
(86)

We now turn our attention to estimating the first term on the right hand of
(86). Recall that according to (70)

$\widehat{W}_{m,n}^{+}(0)\,\leq\,4(1+\varepsilon_{m})(1+\varepsilon_{n})\psi(q_{m})\psi(q_{n})+O\left((q_{m},q_{n})\min\Big{(}\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\Big{)}\right)\,.$

Since $(\varepsilon_{n})_{n\geq a}$ is decreasing and $\varepsilon_{n}\leq
1,\quad n\geq a,$ for all $m,n\in\mathbb{N}$ with $a\leq m<n\leq b,$

$4(1+\varepsilon_{m})(1+\varepsilon_{n})\psi(q_{m})\psi(q_{n})\,\leq\,4\psi(q_{m})\psi(q_{n})+12\varepsilon_{m}\psi(q_{m})\psi(q_{n})\,.$
(87)

Once again, we will consider two cases.

Case 1: $\sum\limits_{k=a}^{b}\psi(q_{k})\leq 1$. It follows, by (80), that
$\varepsilon_{n}=1$ for all $a\leq n\leq b$, therefore

$\mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\varepsilon_{m}\psi(q_{m})\psi(q_{n})\leq\mathop{\sum\sum}\limits_{a\leq
m<n\leq
b}\psi(q_{m})\psi(q_{n})<\left(\sum_{n=a}^{b}\psi(q_{n})\right)^{2}<\sum_{n=a}^{b}\psi(q_{n}).$
(88)

Case 2: $\sum\limits_{k=a}^{b}\psi(q_{k})>1$. Then, by (80),

$\displaystyle\mathop{\sum\sum}\limits_{a\leq m<n\leq b}$
$\displaystyle\varepsilon_{m}\psi(q_{m})\psi(q_{n})=\mathop{\sum\sum}\limits_{a\leq
m<n\leq b}\psi(q_{m})\psi(q_{n})\
\min\left(1,\left(\sum_{k=a}^{m}\psi(q_{k})\right)^{-1}\right)=$ (89)
$\displaystyle=\ \mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\frac{\psi(q_{m})\psi(q_{n})}{\max\left(1,\sum_{k=a}^{m}\psi(q_{k})\right)}\leq\
\sum_{n=a}^{b}\psi(q_{n})\sum_{m=a}^{b}\frac{\psi(q_{m})}{\sum_{k=a}^{m}\psi(q_{k})}\,.$

From Lemma 4 we have that, for all $a,b\in\mathbb{N}$ with $a<b$,

$\sum_{m=a}^{b}\frac{\psi(q_{m})}{\sum_{k=a}^{m}\psi(q_{k})}\leq
1-\log(\psi(q_{a}))+\log\left(\sum_{n=a}^{b}\psi(q_{n})\right).$

This together with (89) implies that

$\displaystyle\mathop{\sum\sum}\limits_{a\leq m<n\leq
b}\varepsilon_{m}\psi(q_{m})\psi(q_{n})$ $\displaystyle\leq\
\left(\sum_{n=a}^{b}\psi(q_{n})\right)\left(1-\log(\psi(q_{a}))+\log\left(\sum_{n=a}^{b}\psi(q_{n})\right)\right)$
(90) $\displaystyle\ll\
\sum_{n=a}^{b}\psi(q_{n})\cdot\log\left(\sum_{n=a}^{b}\psi(q_{n})\right).$

Combining the estimates (70), (87), (88) and (90) we find that

$\mathop{\sum\sum}\limits_{a\leq m<n\leq b}W_{m,n}^{+}(0)\leq
2\left(\sum_{n=a}^{b}\psi(q_{n})\right)^{2}+O\left(\left(\sum_{n=a}^{b}\psi(q_{n})\right)\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)\right)+\\\\[8.61108pt]
+\ O\left(\mathop{\sum\sum}_{a\leq m<n\leq
b}(q_{m},q_{n})\min\Big{(}\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\Big{)}\right)\,,$
(91)

here we used the inequality

$\left(\sum_{n=a}^{b}\psi(q_{n})\right)^{2}=\sum_{n=a}^{b}\psi^{2}(q_{n})+2\mathop{\sum\sum}\limits_{a\leq
m<n\leq b}\psi(q_{m})\psi(q_{n})\geq$ $\geq 2\mathop{\sum\sum}\limits_{a\leq
m<n\leq b}\psi(q_{m})\psi(q_{n}).$

On combining (86) and (91) we find that

$2\mathop{\sum\sum}_{a\leq m<n\leq b}\mu(E_{m}\cap E_{n})\ \leq\
4\left(\sum_{n=a}^{b}\psi(q_{n})\right)^{2}+O\left(\mathop{\sum\sum}_{a\leq
m<n\leq
b}(q_{m},q_{n})\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\right)\\\\[8.61108pt]
+O\left(\left(\sum_{n=a}^{b}\psi(q_{n})\right)\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)+\sum_{n=a}^{b}\psi(q_{n})\right).$

Now we use Lemma 2 to complete the proof of Lemma 5. ∎

###  5.3 Applying Lemma 1 to prove Theorem 1

In this section we will prove our main result, namely Theorem 1. We deduce it
from Lemma 1 using estimates from Lemma 2 and Lemma 5. In order to do this we
will show that all conditions of mentioned above lemmas are satisfied under
conditions of Theorem 1.

###### Proof of Theorem 1.

Note that under balance condition (9) and (10) condition (1) (and, actually,
condition (2) as well) is satisfied. Therefore, the convergence part of
Theorem 1 follows from [17, Theorem 2] Because of this, we need to prove only
the divergence part. So in the rest of the proof we assume

$\Psi(N):=\sum_{n=1}^{N}\psi(q_{n})\to\infty,\quad\text{ when }N\to\infty.$
(92)

Note that all conditions of Lemma 2 are satisfied under assumptions of Theorem
1 and condition (92).

We proceed by showing that it is enough to prove Theorem 1 with the extra
assumption of (64). To this end, introduce two new auxiliary functions
$\omega,\psi^{\ast}:\mathcal{A}\to[0,1]$,

$\omega(q_{n})=n^{-\tau},\quad n\in\mathbb{N},$
$\psi^{\ast}(q_{n})=\max\\{\psi(q_{n}),\omega(q_{n})\\},\quad n\in\mathbb{N}.$

Note that

$\sum_{n=1}^{\infty}\omega(q_{n})=\sum_{n=1}^{\infty}n^{-\tau}<\infty,$

so by convergence part of Theorem 1 already justified in the beginning of the
proof (or see [17, Theorem 2]) we have that counting function
$R(x,N;\gamma,\omega,\mathcal{A})$ remains bounded as $N\to\infty,$ therefore,
in view of Proposition 1 (i), $x\notin W_{\mathcal{A}}(\gamma;\omega).$ So, by
Proposition 1 (iii), we have

$R(x,N;\gamma,\psi^{\ast},\mathcal{A})=R(x,N;\gamma,\psi,\mathcal{A})+O(1).$

This implies that the conclusion of the theorem (11) for $\psi^{*}$ is
equivalent to (11) with original function $\psi$. In the meantime, $\psi^{*}$
obviously satisfies condition (64). So, without loss of generality, we can
assume that $\psi$ satisfies condition (64).

So, we have checked that under conditions of Theorem 1 and assumption (92) all
conditions of Lemma 2 and Lemma 5 are fulfilled, and now we can start to apply
Lemma 1. Using (54) and (65) on the right-hand side of (42), we find that, for
$f_{n}(x)$ and $f_{n}$ defined by (41) and for any $a,b\in\mathbb{N}$ with
$a<b$,

$\displaystyle\int_{\mathbb{I}}\left(\sum_{n=a}^{b}\big{(}f_{n}(x)-f_{n}\big{)}\right)^{2}\mathrm{d}\mu(x)\
\,=\,O$ $\displaystyle\Bigg{(}\
\Big{(}\sum_{n=a}^{b}\psi(q_{n})\Big{)}\Big{(}\log^{+}\Big{(}\sum_{n=a}^{b}\psi(q_{n})\Big{)}+1\Big{)}$
(93) $\displaystyle+\ \mathop{\sum\sum}_{a\leq m<n\leq
b}(q_{m},q_{n})\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\
\Bigg{)}.$

Let $m\in\mathbb{N}$ be the smallest integer satisfying $a\leq m\leq b$ such
that

$\sum_{n=a}^{m}\psi(q_{n})\,\geq\,\frac{1}{2}\sum_{n=a}^{b}\psi(q_{n}).$ (94)

Note that by the definition of $m$, we have that

$\sum_{n=m}^{b}\psi(q_{n})\,\geq\,\frac{1}{2}\sum_{n=a}^{b}\psi(q_{n})$ (95)

and that for any integer $n$ such that $m\leq n\leq b$

$2\Psi(n)\,=2\sum_{k=1}^{n}\psi(q_{k})\,\geq\,\sum_{k=a}^{b}\psi(q_{k}).$ (96)

From inequalities (94), (95) and (96) we have

$\left(\sum_{n=a}^{b}\psi(q_{n})\right)\left(\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)+1\right)\leq\
2\left(\sum_{n=m}^{b}\psi(q_{n})\right)\left(\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)+1\right)\leq$

$\leq\
2\left(\sum_{n=m}^{b}\psi(q_{n})\left(\log^{+}\left(2\Psi(n)\right)+1\right)\right)\leq\
2\left(\sum_{n=m}^{b}\psi(q_{n})\left(\log^{+}\Psi(n)+2\right)\right)\leq$

$\leq\
2\left(\sum_{n=a}^{b}\psi(q_{n})\left(\log^{+}\Psi(n)+2\right)\right)\,.$

Therefore,

$\displaystyle\left(\sum_{n=a}^{b}\psi(q_{n})\right)$
$\displaystyle\left(\log^{+}\left(\sum_{n=a}^{b}\psi(q_{n})\right)+1\right)+$
$\displaystyle+\ \mathop{\sum\sum}_{a\leq m<n\leq
b}(q_{m},q_{n})\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\
\leq\ 2\sum_{n=a}^{b}\phi_{n}\,,$

where

$\phi_{n}:=\psi(q_{n})\left(\log^{+}\Psi(n)+2\right)+\sum_{m=1}^{n-1}(q_{m},q_{n})\min\left(\frac{\psi(q_{m})}{q_{m}},\frac{\psi(q_{n})}{q_{n}}\right)\,.$
(97)

This, together with (93), implies condition (39) of Lemma 1. Now we use Lemma
1 with $X,f_{n}(x)$ and $f_{n}$ given by (41) and $\phi_{n}$ by (97). It is
left to note that for any $n\in\mathbb{N}$, we have that $f_{n}\leq\phi_{n}$,
$f_{n}\leq 2$ and

$\Phi(N):=\sum_{n=1}^{N}\phi_{n}\,\leq\,\Psi(N)\left(\log^{+}\Psi(N)+2\right)\,+\,E(N)\,,$

where $E(N)$ is given by (13). Counting statement (11) now follows from (40).
∎

## References

  * [1] V. Beresnevich, A. Burr,B. Nazer, S. Velani, (eds): Number Theory Meets Wireless Communications, Mathematical Engineering. Springer (2020).
  * [2] V. Beresnevich, S. Velani: _A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures_ , Ann. of Math. (2) 164 (2006), no. 3, 971–992.
  * [3] V. Beresnevich, F. Ramírez, S. Velani: Metric Diophantine approximation: aspects of recent work, in Dynamics and Analytic Number Theory, Editors: Dmitry Badziahin, Alex Gorodnik, and Norbert Peyerimhoff. LMS Lecture Note Series 437, Cambridge University Press, (2016). 1–95.
  * [4] A.S. Besicovitch: Sets of fractional dimensions (IV): on rational approximation to real numbers, J. Lond. Math. Soc. 9 (1934), 126-131.
  * [5] C. Bluhm: Liouville numbers, Rajchman measures, and small Cantor sets, Proceedings of the American Mathematical Society Vol. 128, Numb. 9 (2000), 2637–2640.
  * [6] C. Bluhm: Zur Konstruktion von Salem-Mengen, PhD thesis, University of Erlanger, 1996.
  * [7] C. Bluhm: On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets, Ark. Mat. 36 (1998), no. 2, 307-316.
  * [8] Y. Bugeaud: Nombres de Liouville et nombres normaux, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 117–120.
  * [9] K. Hambrook: Explicit Salem sets and applications to metrical Diophantine approximation, arXiv:1604.00411 (2016)
  * [10] G. Harman: Metric Number Theory. Clarendon Press, Oxford (1998).
  * [11] V. Jarnik, Zur metrischen Theorie der diophantischen Approximation, Prace Mat.-Fiz. 36 (1928-29), 91-106.
  * [12] R. Kaufman: Continued Fractions and Fourier Transforms. Mathematika 27 (1980) 262–267.
  * [13] A. Khintchine: Einige Sätze über Kettenbrüiche mit Anwendugen auf die Theorie der diophantischen Approximationen. Math. Ann. 92 (1924), 115-125.
  * [14] J. Li, T. Sahlsten, Trigonometric series and self-similar sets, J. Eur. Math. Soc. 24(2022), 341–368.
  * [15] P. Mattila: _Geometry of sets and measures in Euclidean spaces._ Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995.
  * [16] P. Mattila: _Fourier analysis and Hausdorff dimension_ , Cambridge Studies in Advanced Mathematics, vol. 150, Cambridge University Press, 2015.
  * [17] A. Pollington, S. Velani, A. Zafeiropoulos, E. Zorin: Inhomogeneous Diophantine Approximation on $M_{0}$-sets with restricted denominators. IMRN Vol. 2022, Issue 11 (2022), 8571–8643.
  * [18] B. P. Rynne,_The Hausdorff dimension of sets arising from Diophantine approximation with a general error function_ , J. Number Theory 71 (1998), no. 2, 166–171.
  * [19] M. Queffélec, O. Ramaré: Analyse de Fourier des fractions continues à quotients restreints. (French) [Fourier analysis of continued fractions with bounded special quotients] Enseign. Math. (2) 49 (2003), no. 3-4, 335-356.
  * [20] W. M. Schmidt: Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110 (1964) 493-518.
  * [21] P. Szüsz: Über die metrische Theorie der diophantischen Approximationen. Acta Math. Acad. Sci. Hungar. 9 (1958), 177-193.

Volodymyr Pavlenkov: Department of Mathematics, University of York,

Volodymyr Pavlenkov: Heslington, York, YO10 5DD, England;

Volodymyr Pavlenkov: Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine.

Volodymyr Pavlenkov: e-mail<EMAIL_ADDRESS>
Evgeniy Zorin: Department of Mathematics, University of York,

Evgeniy Zorin: Heslington, York, YO10 5DD, England.

Evgeniy Zorin: e-mail<EMAIL_ADDRESS>