1 Introduction
Let
$\unicode[STIX]{x1D703}$
be an irrational number. The distribution of the sequence
$\{n\unicode[STIX]{x1D703}\}_{n\geqslant 1}$
in
$\mathbb{R}/\mathbb{Z}$
is a classical topic in Diophantine approximation. It has been studied that
$\{n\unicode[STIX]{x1D703}\}_{n\geqslant 1}$
is well distributed. Weyl’s uniform distribution theorem (e.g. [Reference Kuipers and Niederreiter8]) states that
$\{n\unicode[STIX]{x1D703}\}_{n\geqslant 1}$
is uniformly distributed in
$\mathbb{R}/\mathbb{Z}$
in the sense that for any interval
$(a,b)\subset [0,1)$
,
$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\#\{1\leqslant n\leqslant N:\langle n\unicode[STIX]{x1D703}\rangle \in (a,b)\}=b-a,\end{eqnarray}$$
where
$\langle t\rangle$
for the fractional part of a real number
$t$
. Minkowski’s theorem (e.g. [Reference Minkowski11]) states that if
$y$
is not of the form
$m+\ell \unicode[STIX]{x1D703}$
for integers
$m,\ell$
, then there exist infinitely many integers
$n$
such that
$$\begin{eqnarray}\Vert n\unicode[STIX]{x1D703}-y\Vert <\frac{1}{4|n|},\end{eqnarray}$$
where
$\Vert \cdot \Vert$
for the distance to the nearest integer. Moreover, it was shown by Schmeling and Troubetzkoy [Reference Schmeling and Troubetzkoy12] and by Bugeaud [Reference Bugeaud3] independently (see also [Reference Beresnevich and Velani1]) that for
$\unicode[STIX]{x1D6FE}\geqslant 1$
$$\begin{eqnarray}\dim _{\text{H}}\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <n^{-\unicode[STIX]{x1D6FE}}\text{for infinitely many }n\in \mathbb{N}\}=\frac{1}{\unicode[STIX]{x1D6FE}},\end{eqnarray}$$
where
$\dim _{\text{H}}$
denotes the Hausdorff dimension. These results are all not dependent on the irrational
$\unicode[STIX]{x1D703}$
, but if we replace
$1/n^{\unicode[STIX]{x1D6FE}}$
with a general monotone function
$\unicode[STIX]{x1D711}(n)$
, the situation becomes quite different.
Suppose that
$\unicode[STIX]{x1D711}(n)$
is a positive monotone decreasing function. Let
$$\begin{eqnarray}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <\unicode[STIX]{x1D711}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$
For the Lebesgue measure of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
, Fuchs and Kim [Reference Fuchs and Kim5] showed that
$$\begin{eqnarray}\text{Leb}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))=1\quad \text{if and only if}\quad \mathop{\sum }_{k=0}^{\infty }\mathop{\sum }_{n=q_{k}}^{q_{k+1}-1}\min \{\unicode[STIX]{x1D711}(n),\Vert q_{k}\unicode[STIX]{x1D703}\Vert \}=\infty ,\end{eqnarray}$$
where
$\{q_{k}\}_{k\geqslant 1}$
are the denominators of the principal convergents
$p_{k}/q_{k}$
of
$\unicode[STIX]{x1D703}$
. (For earlier works, see [Reference Bernik and Dodson2, Reference Kim7, Reference Kurzweil9, Reference Tseng13].) This shows that the size of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
depends heavily on the Diophantine properties of
$\unicode[STIX]{x1D703}$
.
For the Hausdorff dimension, Fan and Wu [Reference Fan and Wu4] presented an example indicating that
$\dim _{\text{H}}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
also depends heavily on
$\unicode[STIX]{x1D703}$
. Put
$$\begin{eqnarray}u_{\unicode[STIX]{x1D711}}:=\limsup _{n\rightarrow \infty }\frac{\log n}{-\text{log}\,\unicode[STIX]{x1D711}(n)},\qquad l_{\unicode[STIX]{x1D711}}:=\liminf _{n\rightarrow \infty }\frac{\log n}{-\text{log}\,\unicode[STIX]{x1D711}(n)}.\end{eqnarray}$$
Then, Xu [Reference Xu14] showed that
$$\begin{eqnarray}\limsup _{n\rightarrow \infty }\frac{\log q_{k}}{-\text{log}\,\unicode[STIX]{x1D711}(q_{k})}\leqslant \dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\leqslant u_{\unicode[STIX]{x1D711}}.\end{eqnarray}$$
Moreover, Liao and Rams [Reference Liao and Rams10] proved that
$$\begin{eqnarray}\min \biggl\{u_{\unicode[STIX]{x1D711}},\max \biggl\{l_{\unicode[STIX]{x1D711}},\frac{1+u_{\unicode[STIX]{x1D711}}}{1+w}\biggr\}\biggr\}\leqslant \dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\leqslant u_{\unicode[STIX]{x1D711}},\end{eqnarray}$$
where
$$\begin{eqnarray}w=\limsup _{k\rightarrow \infty }\frac{\log q_{k+1}}{\log q_{k}}.\end{eqnarray}$$
Our main theorem gives the full description of the Hausdorff dimension of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
.
Theorem 1.1. For
$0\leqslant s\leqslant 1$
, let
$q_{k,s}$
be the integer
$m$
with
$q_{k}\leqslant m<q_{k+1}$
minimizing
$$\begin{eqnarray}q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{m-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=m+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}.\end{eqnarray}$$
Then we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\nonumber\\ \displaystyle & & \displaystyle \quad =\inf \biggl\{s>0:\mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr]<\infty \biggr\}.\nonumber\end{eqnarray}$$
Though this statement is complicated, easy calculations lead to the earlier results: we deduce (1.4) by Liao and Rams and (1.3) by Xu from Theorem 1.1 in §4. In §2, we discuss the lower bound of the Hausdorff dimension by the mass transference principle from [Reference Beresnevich and Velani1]. The proof of the main theorem is given in §3.
The proof of the main theorem strongly depends on the result of Liao and Rams. The following proposition, which is going to be crucial in our work, is [Reference Liao and Rams10, Proposition 2.3].
Proposition 1.2 [Reference Liao and Rams10].
Suppose
$K>1$
and
$N>1$
. Let
$\{k_{i}\}$
be a sufficiently fast increasing sequence of integers, for which
$\log q_{k_{i}+1}/\text{log}\,q_{k_{i}}\rightarrow B$
. Let
$\{m_{i}\}$
be a sequence of reals satisfying
$q_{k_{i}}\leqslant m_{i}=q_{k_{i}}^{N}<q_{k_{i}+1}$
. Denote
$$\begin{eqnarray}F_{i}:=\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\dim _{\text{H}}\biggl(\mathop{\bigcap }_{i=1}^{\infty }F_{i}\biggr)=\min \biggl\{\frac{N}{K},\,\max \biggl\{\frac{1}{K},\frac{1}{1+B-N}\biggr\}\biggr\}.\end{eqnarray}$$
We will also need a version of this result which was not proven in [Reference Liao and Rams10], but the proof of which is basically identical.
Proposition 1.3. Let
$\{k_{i}\}$
be a sufficiently fast increasing sequence of integers. Let
$\{K_{i}\}$
and
$\{m_{i}\}$
be two sequences of positive reals satisfying
$K_{i}>1$
and
$q_{k_{i}}\leqslant m_{i}:=q_{k_{i}}^{N_{i}}<q_{k_{i}+1}$
. Write
$\log q_{k_{i}+1}/\text{log}\,q_{k_{i}}=B_{i}$
. Assume that
$K_{i}$
,
$B_{i}-N_{i}$
and
$K_{i}/N_{i}$
converge to
$K,D,R$
respectively. Define
$$\begin{eqnarray}F_{i}:=\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K_{i}}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\dim _{\text{H}}\biggl(\mathop{\bigcap }_{i=1}^{\infty }F_{i}\biggr)=\min \biggl\{\frac{1}{R},\,\max \biggl\{\frac{1}{K},\frac{1}{1+D}\biggr\}\biggr\}.\end{eqnarray}$$
We remark that the exact growth rate of
$\{k_{i}\}_{i\geqslant 1}$
necessary for Propositions 1.2 and 1.3 is not going to be important for us.
2 Mass transference principle
The mass transference principle, developed by Beresnevich and Velani [Reference Beresnevich and Velani1], is a powerful tool to determine the Hausdorff measure and dimension of a lim sup set. For example, equipped with the Minkowski’s result, the dimensional result of (1.1) can be obtained by using the mass transference principle directly. Let
$B(x_{i},r_{i})$
be a sequence of balls in
$[0,1)$
with centers
$x_{i}$
and radii
$r_{i}$
with
$r_{i}\rightarrow 0$
. If
$\limsup B(x_{i},r_{i}^{s})$
has the full Lebesgue measure for
$0<s\leqslant 1$
, then it follows by the mass transference principle [Reference Beresnevich and Velani1] that
$$\begin{eqnarray}\dim _{\text{H}}(\limsup B(x_{i},r_{i}))\geqslant s.\end{eqnarray}$$
Combining (1.2) with the mass transference principle, we have
$$\begin{eqnarray}\dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\geqslant \inf \biggl\{s\geqslant 0:\mathop{\sum }_{k=0}^{\infty }\mathop{\sum }_{n=q_{k}}^{q_{k+1}-1}\min \{\unicode[STIX]{x1D711}(n)^{s},\Vert q_{k}\unicode[STIX]{x1D703}\Vert \}<\infty \biggr\}.\end{eqnarray}$$
However, this may not be the exact dimension of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
.
For each
$0\leqslant s\leqslant 1$
, we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=q_{k}}^{q_{k+1}-1}\min \{\unicode[STIX]{x1D711}(n)^{s},\Vert q_{k}\unicode[STIX]{x1D703}\Vert \} & {\leqslant} & \displaystyle \unicode[STIX]{x1D711}(q_{k})+(q_{k,s}-q_{k})\Vert q_{k}\unicode[STIX]{x1D703}\Vert +\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}.\nonumber\end{eqnarray}$$
Therefore, the lower bound given by (2.1) is less than or equal to the Hausdorff dimension given by Theorem 1.1. However the following example shows that the lower bound may not be sharp.
Example 2.1. Choose an irrational
$\unicode[STIX]{x1D703}$
given by the partial quotients
$a_{k+1}=q_{k}^{2}$
and the error function
$\unicode[STIX]{x1D711}(n)=1/2q_{k}^{3}$
for each
$n\in (q_{k-1}^{2},q_{k}^{2}]$
. By elementary calculus we get
$q_{k,s}=q_{k}^{2}$
. Thus,
$$\begin{eqnarray}\displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s} & {<} & \displaystyle q_{k}^{s+1}\Vert q_{k}\unicode[STIX]{x1D703}\Vert ^{s}+\frac{q_{k+1}}{2^{s}q_{k+1}^{3s}}\nonumber\\ \displaystyle & {<} & \displaystyle \frac{1}{q_{k}^{2s-1}}+\frac{1}{2^{s}q_{k+1}^{3s-1}},\nonumber\end{eqnarray}$$
which converges for
$s>1/2$
. We also have
$$\begin{eqnarray}\displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s} & {>} & \displaystyle q_{k}(q_{k}-1)^{s}\Vert q_{k}\unicode[STIX]{x1D703}\Vert ^{s}\nonumber\\ \displaystyle & {>} & \displaystyle \frac{q_{k}^{1+s}}{2^{3s}q_{k}^{3s}},\nonumber\end{eqnarray}$$
which diverges for
$s<1/2$
. Therefore, the Hausdorff dimension of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
is
$1/2$
.
On the other hand, for
$1/3<s<1$
, we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{q_{k}\leqslant n<q_{k+1}}\min \{\unicode[STIX]{x1D711}(n)^{s},\Vert q_{k}\unicode[STIX]{x1D703}\Vert \} & = & \displaystyle (q_{k}^{2}-q_{k}+1)\Vert q_{k}\unicode[STIX]{x1D703}\Vert +\frac{q_{k+1}-q_{k}^{2}-1}{2q_{k+1}^{3s}}\nonumber\\ \displaystyle & {<} & \displaystyle \frac{1}{q_{k}}+\frac{1}{2q_{k+1}^{3s-1}},\nonumber\end{eqnarray}$$
which is a convergent series. If
$0<s<1/3$
, then for large
$k$
so that
$a_{k+1}=q_{k}^{2}>3$
$$\begin{eqnarray}\mathop{\sum }_{q_{k}\leqslant n<q_{k+1}}\min \{\unicode[STIX]{x1D711}(n)^{s},\Vert q_{k}\unicode[STIX]{x1D703}\Vert \}=(q_{k+1}-q_{k})\Vert q_{k}\unicode[STIX]{x1D703}\Vert >\frac{q_{k+1}-q_{k}}{q_{k+1}+q_{k}}>\frac{1}{2},\end{eqnarray}$$
which is a divergent series. Hence, the lower bound of the Hausdorff dimension given by (2.1) is
$1/3$
.
3 Proof of the main theorem
Let
$\unicode[STIX]{x1D703}=[a_{1},a_{2},\ldots ]$
be the continued fraction expansion of
$\unicode[STIX]{x1D703}$
and
$\{q_{k}\}$
its denominators of the convergents, which satisfy
$q_{k+1}=a_{k+1}q_{k}+q_{k-1}$
. Since
$q_{k+1}\geqslant 2q_{k-1}$
, it increases exponentially. Thus for any
$\unicode[STIX]{x1D700}>0$
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\frac{1}{q_{k}^{\unicode[STIX]{x1D700}}}<\infty .\end{eqnarray}$$
We will use this convergence implicitly.
The Three Distances Theorem (e.g. [Reference Halton6]) plays a central role in the proof. It states that for each positive integer
$N$
, the gaps between two consecutive points of
$\langle \unicode[STIX]{x1D703}\rangle ,\langle 2\unicode[STIX]{x1D703}\rangle ,\ldots ,\langle N\unicode[STIX]{x1D703}\rangle$
have at most three lengths if the points are arranged in ascending order. In the special case, when
$N=q_{k+1}$
the gap between two neighboring points in
$\{\langle \unicode[STIX]{x1D703}\rangle ,\langle 2\unicode[STIX]{x1D703}\rangle ,\ldots ,\langle q_{k+1}\unicode[STIX]{x1D703}\rangle \}$
is
$\Vert q_{k}\unicode[STIX]{x1D703}\Vert$
or
$\Vert q_{k}\unicode[STIX]{x1D703}\Vert +\Vert q_{k+1}\unicode[STIX]{x1D703}\Vert$
. To simplify the notation, we view
$n\unicode[STIX]{x1D703}$
as a number on the unit circle
$S^{1}=\mathbb{R}/\mathbb{Z}$
.
Proof of the upper bound.
To give an upper bound for the Hausdorff dimension of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
, we need to find a suitable cover. We will start by covering not the set
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
itself (which is dense in
$S^{1}$
) but rather the set
$$\begin{eqnarray}E_{k}=\mathop{\bigcup }_{q_{k}\leqslant n<q_{k+1}}B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))\end{eqnarray}$$
for each
$k\geqslant 1$
. Since
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\limsup _{k\rightarrow \infty }E_{k}$
, this will let us construct a cover of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
. Before the main argument, we explain the idea of looking for an optimal cover of
$E_{k}$
.
At first, we arrange the balls
$B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))$
in
$E_{k}$
into
$q_{k}$
groups according to the positions of
$n\unicode[STIX]{x1D703}$
,
$q_{k}\leqslant n<q_{k+1}$
:
$$\begin{eqnarray}\displaystyle \begin{array}{@{}ll@{}}\{B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n)):n=tq_{k}+i,1\leqslant t\leqslant a_{k+1}\} & \text{when }0\leqslant i<q_{k-1};\\ \{B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n)):n=tq_{k}+i,1\leqslant t<a_{k+1}\} & \text{when }q_{k-1}\leqslant i<q_{k}.\end{array} & & \displaystyle\end{eqnarray}$$
Inside each group, it may happen that some balls are overlapping or otherwise sufficiently close so that it will be more efficient to cover several of them with one interval. So, we cover the balls close enough by a long interval, while for other well separated balls, we cover them by themselves. Note that the distances between the centers of neighboring balls are equal (Three Distances Theorem) while their diameters shrink (monotonicity of
$\unicode[STIX]{x1D711}$
), hence we get the following picture: one long interval followed by a sequence of short intervals. This gives a cover of
$E_{k}$
. The role of
$q_{k,s}$
plays is just to optimize the summation of the
$s$
-volume of such a cover.
Now let us give the detailed argument. Let
$0<s\leqslant 1$
be a real number satisfying that
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr]<\infty .\end{eqnarray}$$
For each
$k\geqslant 1$
, inside each group given in (3.1), we cover the balls
$\{B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))\}$
with
$n\leqslant q_{k,s}$
by a long interval. More precisely, let
$c$
,
$r$
be the integers such that
$q_{k,s}=cq_{k}+r$
with
$c\geqslant 1$
,
$0\leqslant r\leqslant q_{k}-1$
. Recall that
$\Vert q_{k}\unicode[STIX]{x1D703}\Vert =(-1)^{k}(q_{k}\unicode[STIX]{x1D703}-p_{k})$
.
If
$c\geqslant 2$
, then let for even
$k$
$$\begin{eqnarray}\displaystyle C_{k,i}:=\left\{\begin{array}{@{}ll@{}}((q_{k}+i)\unicode[STIX]{x1D703}-\unicode[STIX]{x1D711}(q_{k}),(cq_{k}+i)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}(q_{k}))\quad & \text{if }0\leqslant i\leqslant r,\\ ((q_{k}+i)\unicode[STIX]{x1D703}-\unicode[STIX]{x1D711}(q_{k}),((c-1)q_{k}+i)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}(q_{k}))\quad & \text{if }r<i<q_{k},\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
and for odd
$k$
$$\begin{eqnarray}\displaystyle C_{k,i}:=\left\{\begin{array}{@{}ll@{}}((cq_{k}+i)\unicode[STIX]{x1D703}-\unicode[STIX]{x1D711}(q_{k}),(q_{k}+i)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}(q_{k}))\quad & \text{if }0\leqslant i\leqslant r,\\ (((c-1)q_{k}+i)\unicode[STIX]{x1D703}-\unicode[STIX]{x1D711}(q_{k}),(q_{k}+i)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}(q_{k}))\quad & \text{if }r<i<q_{k}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
If
$c=1$
, then for
$0\leqslant i\leqslant r$
let
$$\begin{eqnarray}\displaystyle C_{k,i}:=((q_{k}+i)\unicode[STIX]{x1D703}-\unicode[STIX]{x1D711}(q_{k}),(q_{k}+i)\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}(q_{k})) & & \displaystyle \nonumber\end{eqnarray}$$
and for
$r<i<q_{k}$
let
$C_{k,i}:=\emptyset .$
Then we have
$$\begin{eqnarray}\displaystyle \mathop{\bigcup }_{q_{k}\leqslant n\leqslant q_{k,s}}B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))\subset \mathop{\bigcup }_{q_{k}\leqslant n\leqslant q_{k,s}}B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(q_{k}))\subset \mathop{\bigcup }_{0\leqslant i<q_{k}}C_{k,i}. & & \displaystyle \nonumber\end{eqnarray}$$
As
$$\begin{eqnarray}\displaystyle & & \displaystyle (r+1)(2\unicode[STIX]{x1D711}(q_{k})+(c-1)\Vert q_{k}\unicode[STIX]{x1D703}\Vert )^{s}+(q_{k}-r-1)(2\unicode[STIX]{x1D711}(q_{k})+(c-2)\Vert q_{k}\unicode[STIX]{x1D703}\Vert )^{s}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant q_{k}(2\unicode[STIX]{x1D711}(q_{k})+(c-1)\Vert q_{k}\unicode[STIX]{x1D703}\Vert )^{s}<2q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\biggl(\frac{q_{k,s}}{q_{k}}-1\biggr)\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s},\nonumber\end{eqnarray}$$
we find a covering of the set
$$\begin{eqnarray}\displaystyle E_{k}=\mathop{\bigcup }_{q_{k}\leqslant n<q_{k+1}}B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))\subset \biggl(\mathop{\bigcup }_{0\leqslant i<q_{k}}C_{k,i}\biggr)\cup \biggl(\mathop{\bigcup }_{q_{k,s}<n<q_{k+1}}B(n\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}(n))\biggr) & & \displaystyle \nonumber\end{eqnarray}$$
such that the sum of
$s$
th powers of the lengths of covering intervals is bounded by
$$\begin{eqnarray}2q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+2\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}.\end{eqnarray}$$
As the union over covers of
$E_{k}$
is a cover of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
, it implies that
$\dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\leqslant s$
.◻
In order to prove the lower bound, we use a lemma which is a direct consequence of Proposition 1.3.
Lemma 3.1. Let
$\{k_{i}\}$
be a sufficiently fast increasing sequence of integers. Let
$\{K_{i}\}$
and
$\{m_{i}\}$
be two sequences of positive reals satisfying
$K_{i}>1$
and
$q_{k_{i}}\leqslant m_{i}:=q_{k_{i}}^{N_{i}}<q_{k_{i}+1}$
. Write
$\log q_{k_{i}+1}/\text{log}\,q_{k_{i}}=B_{i}$
. Assume that
$1+B_{i}-N_{i}\leqslant R$
and
$K_{i}/N_{i}\leqslant R$
. Then
$$\begin{eqnarray}\dim _{\text{H}}\biggl(\mathop{\bigcap }_{i=1}^{\infty }\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K_{i}}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}\biggr)\geqslant \frac{1}{R}.\end{eqnarray}$$
Proof of the lower bound.
Suppose that for some
$0<s<1$
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr]=\infty .\end{eqnarray}$$
We have to show that
$\dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\geqslant s-\unicode[STIX]{x1D700}$
for any
$\unicode[STIX]{x1D700}>0$
.
If
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}=\infty , & & \displaystyle \nonumber\end{eqnarray}$$
then either
-
(i)
or
$$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\unicode[STIX]{x1D711}(q_{k})^{s}=\infty & \displaystyle \nonumber\end{eqnarray}$$
-
(ii)
and
$$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\unicode[STIX]{x1D711}(q_{k})^{s}<\infty & \displaystyle \nonumber\end{eqnarray}$$
Otherwise,
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
-
(iii)
and
$$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\\ \displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}<\infty & & \displaystyle \nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl(\mathop{\sum }_{n=q_{k,s}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr)=\infty . & \displaystyle \nonumber\end{eqnarray}$$
Proof of Case (i).
From the condition of Case (i) for each
$\unicode[STIX]{x1D700}>0$
, there exists a subsequence
$\{k_{i}\}$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(q_{k_{i}})>\biggl(\frac{1}{q_{k_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})}.\end{eqnarray}$$
We can freely assume that this subsequence
$\{k_{i}\}$
grows fast enough for us to apply Proposition 1.3 later on. Put
$m_{i}=q_{k_{i}}$
. Then
$$\begin{eqnarray}\unicode[STIX]{x1D711}(m_{i})>\biggl(\frac{1}{q_{k_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})}\geqslant \frac{1}{2q_{k_{i}}^{K}}\quad \text{with }K=\frac{1}{s-\unicode[STIX]{x1D700}}.\end{eqnarray}$$
Let
$$\begin{eqnarray}E_{i}:=\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}.\end{eqnarray}$$
Then the monotonicity of
$\unicode[STIX]{x1D711}$
gives that
$$\begin{eqnarray}E:=\mathop{\bigcap }_{i=1}^{\infty }E_{i}\subset E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})\end{eqnarray}$$
and by Proposition 1.3
$$\begin{eqnarray}\dim _{\text{H}}(E)\geqslant \frac{1}{K}=s-\unicode[STIX]{x1D700}.\end{eqnarray}$$
Proof of Case (ii).
Claim.
For each
$\unicode[STIX]{x1D700}>0$
, there exists a subsequence
$\{k_{i}\}$
such that
$$\begin{eqnarray}\frac{q_{k_{i},s}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert >\max \biggl\{\biggl(\frac{1}{q_{k_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})},\unicode[STIX]{x1D711}(q_{k_{i}})\biggr\}.\end{eqnarray}$$
Proof of the Claim.
Suppose not. Then for all
$k$
sufficiently large
$$\begin{eqnarray}\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \leqslant \max \biggl\{\biggl(\frac{1}{q_{k}}\biggr)^{1/(s-\unicode[STIX]{x1D700})},\unicode[STIX]{x1D711}(q_{k})\biggr\}.\end{eqnarray}$$
Thus
$$\begin{eqnarray}\displaystyle q_{k}\biggl(\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\leqslant \max \biggl\{\biggl(\frac{1}{q_{k}}\biggr)^{\unicode[STIX]{x1D700}/(s-\unicode[STIX]{x1D700})},q_{k}\unicode[STIX]{x1D711}(q_{k})^{s}\biggr\}, & & \displaystyle \nonumber\end{eqnarray}$$
hence
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}<\infty .\square & & \displaystyle \nonumber\end{eqnarray}$$
Let
$m_{i}:=q_{k_{i},s}$
and define
$N_{i},B_{i},K_{i}$
so that
$$\begin{eqnarray}q_{k_{i}}^{N_{i}}=m_{i},\qquad \frac{1}{q_{k_{i}}^{B_{i}}}=\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert ,\qquad \frac{1}{q_{k_{i}}^{K_{i}}}=\unicode[STIX]{x1D711}(m_{i}).\end{eqnarray}$$
Then by the claim
$$\begin{eqnarray}\frac{1}{q_{k_{i}}^{K_{i}}}=\unicode[STIX]{x1D711}(m_{i})\leqslant \unicode[STIX]{x1D711}(q_{k_{i}})<\frac{q_{k_{i},s}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert <\frac{q_{k_{i}+1}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }{q_{k_{i}}}<\frac{1}{q_{k_{i}}}\end{eqnarray}$$
and
$$\begin{eqnarray}q_{k_{i}}^{1+(s-\unicode[STIX]{x1D700})(N_{i}-1-B_{i})}=q_{k_{i}}\biggl(\frac{q_{k_{i},s}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{s-\unicode[STIX]{x1D700}}>q_{k_{i}}\biggl(\frac{q_{k_{i},s}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{s-\unicode[STIX]{x1D700}}>1,\end{eqnarray}$$
which implies
$$\begin{eqnarray}K_{i}>1\end{eqnarray}$$
and
$$\begin{eqnarray}1+B_{i}-N_{i}<\frac{1}{s-\unicode[STIX]{x1D700}}.\end{eqnarray}$$
Set
$$\begin{eqnarray}F:=\mathop{\bigcap }_{i=1}^{\infty }F_{i},\end{eqnarray}$$
where
$$\begin{eqnarray}F_{i}=\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K_{i}}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}.\end{eqnarray}$$
Clearly,
$$\begin{eqnarray}F\subset E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}).\end{eqnarray}$$
If
$q_{k,s}\geqslant q_{k}+1$
, then by the minimality condition of
$q_{k,s}$
, we have
$$\begin{eqnarray}q_{k}\biggl(\!\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \!\biggr)^{s}\leqslant q_{k}\biggl(\!\unicode[STIX]{x1D711}(q_{k})+\frac{(q_{k,s}-1)-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \!\biggr)^{s}+\,\unicode[STIX]{x1D711}(q_{k,s})^{s}.\end{eqnarray}$$
Therefore,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(q_{k,s})^{s} & {\geqslant} & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\nonumber\\ \displaystyle & & \displaystyle -\,q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}-1}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\nonumber\\ \displaystyle & = & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl[1-\biggl(\frac{q_{k}\unicode[STIX]{x1D711}(q_{k})+(q_{k,s}-q_{k}-1)\Vert q_{k}\unicode[STIX]{x1D703}\Vert }{q_{k}\unicode[STIX]{x1D711}(q_{k})+(q_{k,s}-q_{k})\Vert q_{k}\unicode[STIX]{x1D703}\Vert }\biggr)^{s}\biggr]\nonumber\\ \displaystyle & = & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl[1-\biggl(1-\frac{\Vert q_{k}\unicode[STIX]{x1D703}\Vert }{q_{k}\unicode[STIX]{x1D711}(q_{k})+(q_{k,s}-q_{k})\Vert q_{k}\unicode[STIX]{x1D703}\Vert }\biggr)^{s}\biggr]\nonumber\\ \displaystyle & {>} & \displaystyle q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}\cdot \frac{s\Vert q_{k}\unicode[STIX]{x1D703}\Vert }{q_{k}\unicode[STIX]{x1D711}(q_{k})+(q_{k,s}-q_{k})\Vert q_{k}\unicode[STIX]{x1D703}\Vert }\nonumber\\ \displaystyle & = & \displaystyle s\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{-1+s}.\nonumber\end{eqnarray}$$
Here the second inequality is from the fact that
$(1-t)^{s}<1-st$
for
$0<s<1$
and
$0<t<1$
. It is clear from (3.2) that
$q_{k_{i},s}>q_{k_{i}}$
. Therefore, combining the claim (3.2) and the estimation on
$\unicode[STIX]{x1D711}(q_{k,s})$
, we obtain
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(m_{i})^{s}=\unicode[STIX]{x1D711}(q_{k_{i},s})^{s} & {>} & \displaystyle s\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggl(\unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},s}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{-1+s}\nonumber\\ \displaystyle & {>} & \displaystyle s\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggl(\frac{2(q_{k_{i},s}-q_{k_{i}})}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{-1+s}\nonumber\\ \displaystyle & {>} & \displaystyle s\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggl(\frac{2q_{k_{i},s}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{-1+s},\nonumber\end{eqnarray}$$
which is followed by
$$\begin{eqnarray}\displaystyle \frac{1}{q_{k_{i}}^{K_{i}}} & = & \displaystyle \unicode[STIX]{x1D711}(m_{i})>s^{1/s}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert ^{1/s}\biggl(\frac{2m_{i}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{-1/s+1}\nonumber\\ \displaystyle & = & \displaystyle 2\biggl(\frac{s}{2}\biggr)^{1/s}\frac{1}{q_{k_{i}}^{B_{i}}}\biggl(\frac{1}{q_{k_{i}}^{N_{i}-1}}\biggr)^{1/s-1}.\nonumber\end{eqnarray}$$
Thus, for large
$i$
, we obtain
$$\begin{eqnarray}K_{i}-\unicode[STIX]{x1D700}<B_{i}+(N_{i}-1)\biggl(\frac{1}{s}-1\biggr)=1+B_{i}-N_{i}+\frac{1}{s}(N_{i}-1).\end{eqnarray}$$
Applying (3.3) we obtain
$$\begin{eqnarray}K_{i}-\unicode[STIX]{x1D700}<\frac{1}{s-\unicode[STIX]{x1D700}}+\frac{1}{s}(N_{i}-1),\end{eqnarray}$$
thus
$$\begin{eqnarray}\displaystyle \frac{K_{i}}{N_{i}}<\frac{1}{s}+\frac{1}{N_{i}}\biggl(\unicode[STIX]{x1D700}+\frac{1}{s-\unicode[STIX]{x1D700}}-\frac{1}{s}\biggr)\leqslant \unicode[STIX]{x1D700}+\frac{1}{s-\unicode[STIX]{x1D700}}<\frac{1}{s-\unicode[STIX]{x1D700}(1+s^{2})}, & & \displaystyle \nonumber\end{eqnarray}$$
where we apply
$N_{i}\geqslant 1$
for the second inequality. We conclude that
$$\begin{eqnarray}\frac{K_{i}}{N_{i}}<\frac{1}{s-\unicode[STIX]{x1D700}(1+s^{2})}.\end{eqnarray}$$
By (3.3) and (3.4), from Lemma 3.1 we have
$$\begin{eqnarray}\dim _{\text{H}}(F)\geqslant s-\unicode[STIX]{x1D700}(1+s^{2}).\end{eqnarray}$$
Proof of Case (iii).
For each
$\unicode[STIX]{x1D700}>0$
we define a subsequence
$\{q_{k}^{\ast }\}$
by
$$\begin{eqnarray}q_{k}^{\ast }=\left\{\begin{array}{@{}ll@{}}q_{k,s}\quad & \text{if }\unicode[STIX]{x1D6EC}_{k}=\emptyset ,\\ \max \unicode[STIX]{x1D6EC}_{k}\quad & \text{if }\unicode[STIX]{x1D6EC}_{k}\neq \emptyset ,\end{array}\right.\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{k}=\biggl\{q_{k,s}<n<q_{k+1}:\unicode[STIX]{x1D711}(n)\geqslant \frac{1}{n^{1/(s-\unicode[STIX]{x1D700})}}\biggr\}.\end{eqnarray}$$
Then we have
$q_{k,s}\leqslant q_{k}^{\ast }<q_{k+1}$
,
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n)<\frac{1}{n^{1/(s-\unicode[STIX]{x1D700})}}\quad \text{for }q_{k}^{\ast }<n<q_{k+1}\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D711}(q_{k}^{\ast })\geqslant \frac{1}{(q_{k}^{\ast })^{1/(s-\unicode[STIX]{x1D700})}}\quad \text{if }q_{k}^{\ast }>q_{k,s}.\end{eqnarray}$$
By the minimality definition of
$q_{k,s}$
,
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k}^{\ast }-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}+\mathop{\sum }_{n=q_{k}^{\ast }+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr]=\infty .\end{eqnarray}$$
Since
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl(\mathop{\sum }_{n=q_{k}^{\ast }+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{s}\biggr)<\mathop{\sum }_{k=0}^{\infty }\biggl(\mathop{\sum }_{n=q_{k}^{\ast }+1}^{q_{k+1}-1}\frac{1}{n^{1+\unicode[STIX]{x1D700}/(s-\unicode[STIX]{x1D700})}}\biggr)<\mathop{\sum }_{n=1}^{\infty }\frac{1}{n^{1+\unicode[STIX]{x1D700}/(s-\unicode[STIX]{x1D700})}}<\infty ,\end{eqnarray}$$
we obtain
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k}^{\ast }-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
Since
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,s}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}<\infty ,\end{eqnarray}$$
we obtain
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=0}^{\infty }q_{k}\biggl(\frac{q_{k}^{\ast }-q_{k,s}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{s}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
Therefore, we conclude that for each
$\unicode[STIX]{x1D700}>0$
there exists a subsequence
$\{k_{i}\}$
such that
$$\begin{eqnarray}\frac{q_{k_{i}}^{\ast }-q_{k_{i},s}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert >\biggl(\frac{1}{q_{k_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})}.\end{eqnarray}$$
Thus
$$\begin{eqnarray}\frac{q_{k_{i}}^{\ast }}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert >\biggl(\frac{1}{q_{k_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})}\quad \text{and}\quad q_{k_{i}}^{\ast }>q_{k_{i},s}.\end{eqnarray}$$
Put
$$\begin{eqnarray}m_{i}:=q_{k_{i}}^{\ast }=q_{k_{i}}^{N_{i}},\qquad \frac{1}{q_{k_{i}}^{B_{i}}}:=\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert .\end{eqnarray}$$
On one hand, by (3.7),
$$\begin{eqnarray}q_{k_{i}}\biggl(\frac{q_{k_{i}}^{\ast }}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{s-\unicode[STIX]{x1D700}}=q_{k_{i}}^{1-(s-\unicode[STIX]{x1D700})(1+B_{i}-N_{i})}>1,\end{eqnarray}$$
which implies
$$\begin{eqnarray}1+B_{i}-N_{i}<\frac{1}{s-\unicode[STIX]{x1D700}}.\end{eqnarray}$$
On the other hand, since
$q_{k_{i}}^{\ast }>q_{k_{i},s}$
(see (3.7)), by (3.6), one has that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(m_{i})=\unicode[STIX]{x1D711}(q_{k_{i}}^{\ast })\geqslant \biggl(\frac{1}{q_{k_{i}}^{\ast }}\biggr)^{1/(s-\unicode[STIX]{x1D700})}=\biggl(\frac{1}{q_{k_{i}}^{N_{i}}}\biggr)^{1/(s-\unicode[STIX]{x1D700})}.\end{eqnarray}$$
Set
$$\begin{eqnarray}K_{i}:=\frac{N_{i}}{s-\unicode[STIX]{x1D700}}>1\end{eqnarray}$$
and define
$$\begin{eqnarray}G_{i}:=\biggl\{y:\Vert n\unicode[STIX]{x1D6FC}-y\Vert <\frac{1}{2q_{k_{i}}^{K_{i}}}\text{ for some }m_{i-1}<n\leqslant m_{i}\biggr\}.\end{eqnarray}$$
It is clear that
$$\begin{eqnarray}G:=\mathop{\bigcap }_{i=1}^{\infty }G_{i}\subset E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}).\end{eqnarray}$$
Therefore, by (3.8), (3.9) and Lemma 3.1
$$\begin{eqnarray}\displaystyle \dim _{\text{H}}(E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703}))\geqslant \dim _{\text{H}}(G)\geqslant s-\unicode[STIX]{x1D700}.\square & & \displaystyle \nonumber\end{eqnarray}$$
4 Bounds for the dimension
In this section, we show that the previous known dimensional results are deduced by the main theorem.
Corollary 4.1 (Schmeling–Troubetzkoy [Reference Schmeling and Troubetzkoy12], Bugeaud [Reference Bugeaud3]).
We have
$$\begin{eqnarray}l_{\unicode[STIX]{x1D711}}\leqslant \dim _{\text{H}}(E)\leqslant u_{\unicode[STIX]{x1D711}}.\end{eqnarray}$$
Proof. (i) By the definition of
$l_{\unicode[STIX]{x1D711}}$
, for any
$\unicode[STIX]{x1D700}>0$
for sufficiently large
$n$
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n)>\frac{1}{n^{1/(l-\unicode[STIX]{x1D700})}}.\end{eqnarray}$$
Therefore, for some large
$k_{0}$
we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,l-\unicode[STIX]{x1D700}}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{l-\unicode[STIX]{x1D700}}+\mathop{\sum }_{n=q_{k,l-\unicode[STIX]{x1D700}}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{l-\unicode[STIX]{x1D700}}\biggr]\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \mathop{\sum }_{k=k_{0}}^{\infty }q_{k}\unicode[STIX]{x1D711}(q_{k})^{l-\unicode[STIX]{x1D700}}\nonumber\\ \displaystyle & & \displaystyle \quad >\mathop{\sum }_{k=0}^{\infty }1=\infty .\nonumber\end{eqnarray}$$
Hence, we have
$$\begin{eqnarray}l_{\unicode[STIX]{x1D711}}\leqslant \dim _{\text{H}}(E).\end{eqnarray}$$
(ii) By the definition of
$u_{\unicode[STIX]{x1D711}}$
, for any
$\unicode[STIX]{x1D70C}>1$
for sufficiently large
$n$
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n)<\frac{1}{n^{1/\unicode[STIX]{x1D70C}u}}.\end{eqnarray}$$
Therefore, by the minimality of the definition of
$q_{k,\unicode[STIX]{x1D70C}^{2}u}$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,\unicode[STIX]{x1D70C}^{2}u}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{\unicode[STIX]{x1D70C}^{2}u}+\mathop{\sum }_{n=q_{k,\unicode[STIX]{x1D70C}^{2}u}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{\unicode[STIX]{x1D70C}^{2}u}\biggr]\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathop{\sum }_{k=0}^{\infty }\biggl(q_{k}\unicode[STIX]{x1D711}(q_{k})^{\unicode[STIX]{x1D70C}^{2}u}+\mathop{\sum }_{n=q_{k}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{\unicode[STIX]{x1D70C}^{2}u}\biggr)\leqslant \mathop{\sum }_{k=0}^{\infty }\biggl(\frac{1}{q_{k}^{\unicode[STIX]{x1D70C}-1}}+\mathop{\sum }_{n=q_{k}}^{q_{k+1}-1}\frac{1}{n^{\unicode[STIX]{x1D70C}}}\biggr)<\infty ,\nonumber\end{eqnarray}$$
which implies that
$$\begin{eqnarray}\displaystyle \dim _{\text{H}}(E)\leqslant \unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D711}}.\square & & \displaystyle \nonumber\end{eqnarray}$$
Corollary 4.2 (Xu [Reference Xu14]).
We have
$$\begin{eqnarray}\dim _{\text{H}}(E)\geqslant \limsup _{k\rightarrow \infty }\frac{\log q_{k}}{-\text{log}\,\unicode[STIX]{x1D711}(q_{k})}.\end{eqnarray}$$
Proof. Let
$$\begin{eqnarray}t:=\limsup _{k\rightarrow \infty }\frac{\log q_{k}}{-\text{log}\,\unicode[STIX]{x1D711}(q_{k})}.\end{eqnarray}$$
Then for any
$\unicode[STIX]{x1D700}>0$
there exists
$q_{k_{i}}$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(q_{k_{i}})>\frac{1}{q_{k_{i}}^{1/(t-\unicode[STIX]{x1D700})}}.\end{eqnarray}$$
Hence, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl(q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,t-\unicode[STIX]{x1D700}}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{t-\unicode[STIX]{x1D700}}+\mathop{\sum }_{n=q_{k,t-\unicode[STIX]{x1D700}}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{t-\unicode[STIX]{x1D700}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \mathop{\sum }_{i=1}^{\infty }q_{k_{i}}\unicode[STIX]{x1D711}(q_{k_{i}})^{t-\unicode[STIX]{x1D700}}\nonumber\\ \displaystyle & & \displaystyle \quad >\mathop{\sum }_{i=1}^{\infty }1=\infty ,\nonumber\end{eqnarray}$$
which implies that
$$\begin{eqnarray}\displaystyle t\leqslant \dim _{\text{H}}(E).\square & & \displaystyle \nonumber\end{eqnarray}$$
Corollary 4.3 (Liao and Rams [Reference Liao and Rams10]).
If
$(1+u_{\unicode[STIX]{x1D711}})/(1+w)<u_{\unicode[STIX]{x1D711}}$
, then
$$\begin{eqnarray}\dim _{\text{H}}(E)\geqslant \frac{1+u_{\unicode[STIX]{x1D711}}}{1+w}.\end{eqnarray}$$
Proof. Write
$u_{\unicode[STIX]{x1D711}}=u$
. Let
$$\begin{eqnarray}z:=\frac{1+u}{1+w}<u\end{eqnarray}$$
and
$0<\unicode[STIX]{x1D700}<z$
be given. Since
$\limsup -\text{log}\Vert q_{k}\unicode[STIX]{x1D703}\Vert /\text{log}\,q_{k}=w$
, for all
$n$
large,
$$\begin{eqnarray}\Vert q_{n}\unicode[STIX]{x1D703}\Vert >\frac{1}{q_{n}^{w+\unicode[STIX]{x1D700}}}.\end{eqnarray}$$
There are two cases as follows.
-
(a) If there are infinitely many
$n$
such that
$q_{k}\leqslant n\leqslant q_{k,z-\unicode[STIX]{x1D700}}$
and
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n)>\frac{1}{n^{1/(u-\unicode[STIX]{x1D700})}}.\end{eqnarray}$$
-
(b) Suppose that for all large
$n$
such that
$q_{k}\leqslant n\leqslant q_{k,z-\unicode[STIX]{x1D700}}$
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n)\leqslant \frac{1}{n^{1/(u-\unicode[STIX]{x1D700})}}.\end{eqnarray}$$
Case (a). There exist
$\{n_{i}\}$
and
$\{k_{i}\}$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n_{i})>\frac{1}{n_{i}^{1/(u-\unicode[STIX]{x1D700})}},\qquad q_{k_{i}}\leqslant n_{i}\leqslant q_{k_{i},z-\unicode[STIX]{x1D700}}.\end{eqnarray}$$
Then we have
$$\begin{eqnarray}\unicode[STIX]{x1D711}(q_{k_{i}})\geqslant \unicode[STIX]{x1D711}(n_{i})>\frac{1}{n_{i}^{1/(u-\unicode[STIX]{x1D700})}}\geqslant \frac{1}{(q_{k_{i},z-\unicode[STIX]{x1D700}})^{1/(u-\unicode[STIX]{x1D700})}}.\end{eqnarray}$$
Thus, since
$q_{k_{i},z-\unicode[STIX]{x1D700}}\geqslant q_{k_{i}}$
, we obtain
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert >\frac{1}{(q_{k_{i},z-\unicode[STIX]{x1D700}})^{1/(u-\unicode[STIX]{x1D700})}}+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \geqslant \min _{t\geqslant q_{k_{i}}}f_{u-\unicode[STIX]{x1D700}}(t), & & \displaystyle \nonumber\end{eqnarray}$$
where we set
$f_{v}(t):=1/t^{1/v}+t(\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert /q_{k_{i}})$
.
By elementary calculus, we have
$$\begin{eqnarray}\min _{t\geqslant t_{0}}f_{v}(t)=\left\{\begin{array}{@{}ll@{}}\displaystyle f_{v}\biggl(\biggl(\frac{q_{k_{i}}}{v\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }\biggr)^{v/(1+v)}\biggr)\quad & \displaystyle \text{if }0<t_{0}\leqslant \biggl(\frac{q_{k_{i}}}{v\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }\biggr)^{v/(1+v)},\\ f_{v}(t_{0})\quad & \displaystyle \text{if }t_{0}>\biggl(\frac{q_{k_{i}}}{v\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }\biggr)^{v/(1+v)}.\end{array}\right.\end{eqnarray}$$
Therefore, if
$q_{k_{i}}\leqslant (q_{k_{i}}/(u-\unicode[STIX]{x1D700})\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert )^{(u-\unicode[STIX]{x1D700})/(1+u-\unicode[STIX]{x1D700})}$
, then
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert & {>} & \displaystyle \biggl(\frac{(u-\unicode[STIX]{x1D700})\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }{q_{k_{i}}}\biggr)^{1/(1+u-\unicode[STIX]{x1D700})}\nonumber\\ \displaystyle & & \displaystyle +\,\biggl(\frac{q_{k_{i}}}{(u-\unicode[STIX]{x1D700})\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }\biggr)^{(u-\unicode[STIX]{x1D700})/(1+u-\unicode[STIX]{x1D700})}\frac{\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert }{q_{k_{i}}}\nonumber\\ \displaystyle & {>} & \displaystyle (u-\unicode[STIX]{x1D700})^{1/(1+u-\unicode[STIX]{x1D700})}\biggl(\frac{1}{q_{k_{i}}}\biggr)^{(1+w+\unicode[STIX]{x1D700})/(1+u-\unicode[STIX]{x1D700})}+\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \nonumber\\ \displaystyle & {>} & \displaystyle \biggl(\frac{u-\unicode[STIX]{x1D700}}{q_{k_{i}}}\biggr)^{1/(z-\unicode[STIX]{x1D700})}+\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \nonumber\end{eqnarray}$$
and if
$q_{k_{i}}>(q_{k_{i}}/(u-\unicode[STIX]{x1D700})\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert )^{(u-\unicode[STIX]{x1D700})/(1+u-\unicode[STIX]{x1D700})}$
, then
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert >\frac{1}{(q_{k_{i}})^{1/(u-\unicode[STIX]{x1D700})}}+\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \geqslant \frac{1}{(q_{k_{i}})^{1/(z-\unicode[STIX]{x1D700})}}+\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert . & & \displaystyle \nonumber\end{eqnarray}$$
Thus,
$$\begin{eqnarray}\displaystyle q_{k_{i}}\biggl(\unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{z-\unicode[STIX]{x1D700}}>u-\unicode[STIX]{x1D700}. & & \displaystyle \nonumber\end{eqnarray}$$
Hence, we have
$$\begin{eqnarray}\mathop{\sum }_{i=1}^{\infty }\biggl[q_{k_{i}}\biggl(\unicode[STIX]{x1D711}(q_{k_{i}})+\frac{q_{k_{i},z-\unicode[STIX]{x1D700}}-q_{k_{i}}}{q_{k_{i}}}\Vert q_{k_{i}}\unicode[STIX]{x1D703}\Vert \biggr)^{z-\unicode[STIX]{x1D700}}+\mathop{\sum }_{n=q_{k_{i},z-\unicode[STIX]{x1D700}}+1}^{q_{k_{i}+1}-1}\unicode[STIX]{x1D711}(n)^{z-\unicode[STIX]{x1D700}}\biggr]=\infty ,\end{eqnarray}$$
which implies that
$$\begin{eqnarray}z-\unicode[STIX]{x1D700}\leqslant \dim _{\text{H}}(E).\end{eqnarray}$$
Case (b). By the definition of
$u=\limsup (\log n/-\text{log}\,\unicode[STIX]{x1D711}(n))$
, there exist
$\{n_{i}\}$
and
$\{k_{i}\}$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D711}(n_{i})>\frac{1}{n_{i}^{1/(u-\unicode[STIX]{x1D700}/2)}},\qquad q_{k_{i},z-\unicode[STIX]{x1D700}}<n_{i}<q_{k_{i}+1}.\end{eqnarray}$$
Then for large
$i$
we obtain
$$\begin{eqnarray}\frac{1}{(q_{k_{i},z-\unicode[STIX]{x1D700}})^{1/(u-\unicode[STIX]{x1D700})}}\geqslant \unicode[STIX]{x1D711}(q_{k_{i},z-\unicode[STIX]{x1D700}})\geqslant \unicode[STIX]{x1D711}(n_{i})>\frac{1}{(n_{i})^{1/(u-\unicode[STIX]{x1D700}/2)}}.\end{eqnarray}$$
Thus, for large
$i$
we have
$$\begin{eqnarray}q_{k_{i},z-\unicode[STIX]{x1D700}}<(n_{i})^{(u-\unicode[STIX]{x1D700})/(u-\unicode[STIX]{x1D700}/2)}<\frac{n_{i}}{2}.\end{eqnarray}$$
Therefore, for large
$i$
we have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=q_{k_{i},z-\unicode[STIX]{x1D700}}+1}^{q_{k_{i}+1}-1}\unicode[STIX]{x1D711}(n)^{z-\unicode[STIX]{x1D700}} & {\geqslant} & \displaystyle \mathop{\sum }_{n=q_{k_{i},z-\unicode[STIX]{x1D700}}+1}^{q_{k_{i}+1}-1}\unicode[STIX]{x1D711}(n)^{u-\unicode[STIX]{x1D700}}\geqslant \mathop{\sum }_{n=q_{k_{i},z-\unicode[STIX]{x1D700}}+1}^{n_{i}}\unicode[STIX]{x1D711}(n_{i})^{u-\unicode[STIX]{x1D700}}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \frac{n_{i}-q_{k_{i},z-\unicode[STIX]{x1D700}}}{(n_{i})^{(u-\unicode[STIX]{x1D700})/(u-\unicode[STIX]{x1D700}/2)}}>1.\nonumber\end{eqnarray}$$
Hence, we have
$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl[q_{k}\biggl(\unicode[STIX]{x1D711}(q_{k})+\frac{q_{k,z-\unicode[STIX]{x1D700}}-q_{k}}{q_{k}}\Vert q_{k}\unicode[STIX]{x1D703}\Vert \biggr)^{z-\unicode[STIX]{x1D700}}+\mathop{\sum }_{n=q_{k,z-\unicode[STIX]{x1D700}}+1}^{q_{k+1}-1}\unicode[STIX]{x1D711}(n)^{z-\unicode[STIX]{x1D700}}\biggr]=\infty ,\end{eqnarray}$$
which implies that
$$\begin{eqnarray}\displaystyle z-\unicode[STIX]{x1D700}\leqslant \dim _{\text{H}}(E).\square & & \displaystyle \nonumber\end{eqnarray}$$
Acknowledgements
The authors would like to thank Huazhong University of Science and Technology for inviting two of the authors. D.K. is supported by NRF-2015R1A2A2A01007090. M.R. is supported by National Science Centre grant 2014/13/B/ST1/01033 (Poland). B.W. is supported by NSFC 11471130 and 11722105.
