2. Preliminaries

In this section, we recall the notion of the Hausdorff dimension and the summation method.

2.1 Hausdorff dimension

For a non-empty subset $F\subset \mathbb R^n$ and $s\ge 0$ , we define the s-dimensional Hausdorff measure of F as follows [Reference Falconer8, Section 2.1]:

\begin{equation*}\mathcal H^s(F) \;:\!=\; \lim_{\delta\to0} \inf \left\{\sum_{i=1}^\infty \left({\rm diam} \ U_i \right)^s \;:\; F\subset \bigcup_{i\;=\;1}^\infty U_i, \ 0\le {\rm diam} \ U_i \le \delta \right\}.\end{equation*}

The Hausdorff dimension of the set $F\subset \mathbb R^n$ is defined as follows [Reference Falconer8, Section 2.2]:

\begin{equation*}{\rm dim}_H F \;:\!=\; \inf\{ s\ge 0 \;:\; \mathcal H^s(F)\;=\;0\}.\end{equation*}

2.2 Methods of summation

Any matrix $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ generates a (matrix) method of summation given by the natural mapping

\begin{equation*}x\mapsto \left\{\sum_{k\;=\;1}^\infty a_{nk}x_k\right\}_{n\;=\;1}^\infty,\end{equation*}

on the space of all sequences $x\;=\;\{x_k\}_{k\;=\;1}^\infty$ . Whereas this definition is sufficient for our purposes, we refer to [Reference Boos3, Definition 1.2.12] for a general definition of the summation method.

Definition 2.1. A (matrix) method of summation $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ is said to be

1. – regular if for every x such that $x_n\to c$ as $n\to\infty$ , one has

   \begin{equation*}\sum_{k\;=\;1}^\infty a_{nk}x_k\to c \ \textrm{as} \ n\to\infty.\end{equation*}

2. – strongly regular if

   \begin{equation*}\sum_{k\;=\;1}^\infty |a_{nk}-a_{n,k\;+\;1}|\to 0 \ \textrm{as} \ n\to\infty.\end{equation*}

Definition 2.2. Let $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ and $B\;=\;\{b_{nk}\}_{n,k\;=\;1}^\infty$ be two (matrix) methods of summation.

1. 1. $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ is said to be weaker than $B\;=\;\{b_{nk}\}_{n,k\;=\;1}^\infty$ if the existence of the limit

   \begin{equation*}\lim_{n\to\infty}\sum_{k\;=\;1}^\infty a_{nk}x_k,\end{equation*}
   implies the existence of the limit
   \begin{equation*}\lim_{n\to\infty}\sum_{k\;=\;1}^\infty b_{nk}x_k,\end{equation*}
   for every x;

2. 2. $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ is said to be consistent with $B\;=\;\{b_{nk}\}_{n,k\;=\;1}^\infty$ if the existence of both limits

   \begin{equation*}\lim_{n\to\infty}\sum_{k\;=\;1}^\infty a_{nk}x_k \ \text{and } \ \lim_{n\to\infty}\sum_{k\;=\;1}^\infty b_{nk}x_k,\end{equation*}
   implies that their values are the same for every x.

3. Main result

For every $m, p, q\in{\mathbb N}$ such that $0\le p \le q$ consider sets

\begin{equation*}A_{m,p,q} \;:\!=\; \left\{ t\in [0,1] \;:\; \sum_{i\;=\;qnm+1}^{qnm+qm}x_i(t)\;=\;p\cdot m \ \text{for every} \ n\in {\mathbb N}\right\}.\end{equation*}

In words, the sets $A_{m,p,q}$ consist of all $t\in [0,1]$ such that in their binary representation every block of the form $[qnm+1, qnm+qm]$ contains exactly pm ones.

For every fixed $p, q\in{\mathbb N}$ such that $0\le p \le q$ the inclusion $A_{m,p,q} \subset G^{p/q}$ holds for every $m\in {\mathbb N}$ . Indeed, for every large enough $n\in {\mathbb N}$ , there exists $i \in {\mathbb N}$ such that $qmi < n \le qm(i+1)$ . Then, regardless of $j\in {\mathbb N}$ , for every $t\in A_{m,p,q}$ we have

\begin{equation*}pm (i-1) <\sum\limits_{k\;=\;j}^{j\;+\;n\;-\;1} x_k(t) \le pm (i\;+\;2).\end{equation*}

Dividing by n and taking the limit as $n\to\infty$ (or, equivalently, as $i\to\infty$ ), we obtain

\begin{equation*} \lim\limits_{n\to\infty} \frac1n \sum\limits_{k\;=\;j}^{j\;+\;n\;-\;1} x_k(t)\;=\;\frac{p}{q},\end{equation*}

uniformly in $j\in{\mathbb N}$ . Thus, $x(t) \in ac_\alpha$ with $\alpha\;=\;p/q$ .

Lemma 3.1. For every $m, p, q\in{\mathbb N}$ such that $0\le p \le q$ the Hausdorff dimension of the set $A_{m,p,q}$ is

\begin{equation*}{\rm dim}_H (A_{m,p,q})=\frac{\log_2 C_{qm}^{pm}}{qm},\end{equation*}

where $C_n^k$ are binomial coefficients.

Proof. Let $w_j$ , $j=1,\dots, C_{qm}^{pm}$ be all words (on the alphabet $\{0,1\}$ ) of length qm, which contain exactly pm ones. For every $j=1,\dots, C_{qm}^{pm}$ consider intervals $I_j=[a_j,b_j]\subset [0,1]$ , where $a_j=0.w_j00\dots_2$ , $b_j=0.w_j11\dots_2$ . Here, the subscript 2 means the binary representation as in the formula (1.1).

For every $j=1,\dots, C_{qm}^{pm}$ define functions $f_j \;:\; [0,1] \to I_j$ as follows:

\begin{equation*}f_j(t)\;=\;a_j\;+\;(b_j\;-\;a_j)t.\end{equation*}

Using representation (1.1) for $b_j-a_j$ and t and the Cauchy product formula, we obtain that

\begin{equation*}(b_j\;-\;a_j)t\;=\;0.\underbrace{0...0}_{qm}x_1(t)x_2(t)x_3(t)\dots_2.\end{equation*}

Thus, $f_j(t)=0.w_j x_1(t)x_2(t)x_3(t)\dots_2$ and so,

\begin{equation*}A_{m,p,q}\;=\;\bigcup_{j\;=\;1}^{C_{qm}^{pm}} f_j(A_{m,p,q}).\end{equation*}

This tells us that the set $A_{m,p,q}$ is an attractor of the iterated function system $\{f_1,\dots, f_{C_{qm}^{pm}}\}$ .

The length of all intervals $I_j$ is

\begin{equation*}|I_j|\;=\;0.w_j11\dots_2-0.w_j00\dots_2\;=\;0.\underbrace{0...0}_{qm}11\dots_2\;=\;0.\underbrace{0...0}_{qm-1}100\dots_2\;=\;2^{-qm}.\end{equation*}

Since all $f_j$ ’s are linear functions from [0,Reference Albeverio, Pratsiovytyi and Torbin1] to $I_j$ , it follows that all $f_j$ are similarities with ratios $r_j=2^{-qm}$ for all $j=1,\dots, C_{qm}^{pm}$ .

By construction the intervals $I_j$ and $I_k$ intersect if and only if $w_j=\beta01$ and $w_k\;=\;\beta10$ for some word $\beta$ . Indeed, we have then $a_k\;=\;0.\beta100\dots_2$ and $b_j=0.\beta011\dots_2$ . So, the point $0.\beta100\dots_2$ belongs to both intervals. Thus, the open intervals $(a_j,b_j)$ , $j=1,\dots, C_{qm}^{pm}$ are pairwise disjoint. Therefore, the sets $f_j(0,1)$ , $j=1,\dots, C_{qm}^{pm}$ are pairwise disjoint and

\begin{equation*}\bigcup_{j\;=\;1}^{C_{qm}^{pm}} f_j(0,1) \subset (0,1).\end{equation*}

This means, that the iterated function system $\{f_1,\dots, f_{C_{qm}^{pm}}\}$ satisfies the open set condition (see e.g. [Reference Falconer8, p. 129]).

By [Reference Falconer8, Theorem 9.3] the Hausdorff dimension d of the set $A_{m,p,q}$ is a solution of the following equation:

\begin{equation*}\sum_{j\;=\;1}^{C_{qm}^{pm}} r_j^d\;=\;1.\end{equation*}

Since all $r_j=2^{-qm}$ , it follows that

\begin{equation*}d\;=\;\frac{\log_2 C_{qm}^{pm}}{qm},\end{equation*}

as required.

Theorem 3.2. Let $\alpha\;=\;p/q$ with $p, q\in{\mathbb N}$ such that $0\le p \le q$ . The Hausdorff dimension of the set $G^\alpha$ is

\begin{equation*}-\alpha \log_2 \alpha\;-\;(1-\alpha) \log_2 (1-\alpha).\end{equation*}

Proof. Since $A_{m,p,q} \subset G^\alpha$ for every $m\in {\mathbb N}$ , it follows that

\begin{equation*}{\rm dim}_H (G^\alpha)\ge \lim_{m\to\infty}{\rm dim}_H (A_{m,p,q})\;=\;\lim_{m\to\infty} \frac{\log_2 C_{qm}^{pm}}{qm}.\end{equation*}

Using the Stirling formula, we obtain

\begin{align*}{\rm dim}_H (G^\alpha)&\ge \lim_{m\to\infty} \frac{1}{qm}\log_2\frac{\sqrt{2\pi qm} \left(\frac{qm}{e}\right)^{qm}}{\sqrt{2\pi pm} \left(\frac{pm}{e}\right)^{pm}\cdot \sqrt{2\pi (q-p)m} \left(\frac{(q-p)m}{e}\right)^{(q-p)m}}\\&=\lim_{m\to\infty} \frac{1}{qm}\log_2\left[\sqrt{\frac{1}{2\pi m}\frac{q}{p(q-p)}}\cdot \left(\frac{q}{q-p} \right)^{qm}\cdot \left(\frac{q-p}{p} \right)^{pm}\right]\\&=\log_2\frac{q}{q-p}\;+\;\frac{p}{q}\log_2\frac{q-p}{p}\\&=\log_2\frac{1}{1-\alpha}\;+\;\alpha\log_2\frac{1-\alpha}{\alpha}\\&=-\alpha \log_2 \alpha\;-\;(1-\alpha) \log_2 (1-\alpha).\end{align*}

To prove the upper bound, we note that it follows directly from the definition that every almost convergent sequence is Cesàro convergent. Thus, $G^\alpha \subset F^\alpha$ for every $0\le \alpha \le 1$ and

\begin{equation*}{\rm dim}_H (G^\alpha)\le {\rm dim}_H (F^\alpha)\;=\;-\alpha \log_2 \alpha\;-\;(1-\alpha) \log_2 (1-\alpha),\end{equation*}

by the aforementioned Besicovitch result. These two estimates prove the assertion.

Consider the sequence

\begin{equation*}x\;=\;\sum_{i\;=\;1}^\infty \chi_{[r_im\;+\;1, r_im\;+\;p_im]},\end{equation*}

where $\chi$ stands for the characteristic function of a set. It is easy to see that $x\in A_{m,\alpha, \textbf{p},\textbf{q}}$ . One the other hand, since $\alpha$ is irrational, it follows that both sequences $\{p_i\}$ and $\{q_i-p_i\}$ tend to infinity. Thus, the sequence x is not almost convergent. Hence, the inclusion $A_{m,\alpha, \textbf{p},\textbf{q}} \subset G^\alpha$ fails.

Proof. Since ${\rm dim}_H (G^{1/2})=1$ , it follows that 1-dimensional Hausdorff measure of the set $G^{1/2}$ equal to its Lebesgue measure and equal to zero by [Reference Connor6].

The following two results compute the dimension of the level sets for a wide class of summation methods.

Corollary 3.5. Let $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ be a strongly regular summation method, which is weaker than (and consistent with) the Cesàro method, and let $0\le \alpha\le 1$ be a rational number. The Hausdorff dimension of the set

\begin{equation*}\left\{ t\in [0,1] \;:\; \lim_{n\to\infty}\sum_{k\;=\;1}^\infty a_{nk}x_k(t)\;=\;\alpha\right\},\end{equation*}

is

\begin{equation*}-\alpha \log_2 \alpha\;-\;(1-\alpha) \log_2 (1-\alpha).\end{equation*}

Proof. In [Reference Lorentz9, Theorem 7] G. G. Lorentz showed that every strongly regular summation method sums every almost convergent sequence (to the same value). On the other hand, since A is weaker than the Cesàro method, every A-summable sequence is Cesàro summable. Therefore,

(3.1) \begin{equation}G^\alpha \subseteq\left\{ t\in [0,1] \;:\; \lim_{n\to\infty}\frac{1}{n}\sum_{k\;=\;1}^n a_{nk}x_k(t)\;=\;\alpha\right\}\subseteq F^\alpha.\end{equation}

The assertion follows from Theorem 3.2 and the above-mentioned Besicovitch result.

Corollary 3.6. Let $A\;=\;\{a_{nk}\}_{n,k\;=\;1}^\infty$ be a strongly regular summation method. The Hausdorff dimension of the set

\begin{equation*}\left\{ t\in [0,1] \;:\; \lim_{n\to\infty}\sum_{k\;=\;1}^\infty a_{nk}x_k(t)\;=\;\frac12\right\},\end{equation*}

is $1.$

Proof. The assertion follows from the first inclusion in (3.1) and Corollary 3.4.

To demonstrate that Corollaries 3.5 and 3.6, in fact, hold for a broad range of summation methods, we verify the assertions for families of the general Cesàro and Riesz summation methods.

Example 3.7. For $\beta>0$ consider the Cesàro method of summation $C^\beta\;=\;\{c^{(\beta)}_{nk}\}_{n,k\;=\;1}^\infty$ , where

\begin{equation*}c^{(\beta)}_{nk}\;=\;\frac{\binom{ n-k\;+\;\beta -1}{ n-k}}{ \binom{n\;+\;\beta }{ n}}, \ \text{for} \ k\le n,\end{equation*}

and zero otherwise.

For every $\beta>0$ , the method of summation $C^\beta$ satisfies Corollary 3.6 by [Reference Lorentz9, p. 179]. For every $0<\beta\le 1$ , the method of summation $C^\beta$ satisfies Corollary 3.5 by [Reference Boos3, Theorem 3.1.10].

Example 3.8. For a fixed sequence $\{p_n\}_{n\;=\;1}^\infty$ of real numbers such that $p_0>0$ and $p_n\ge0$ set $P_n\;=\;\sum\nolimits_{k\;=\;1}^n p_k\Phi$ and consider the Riesz method of summation $R_p=\{r_{nk}\}_{n,k\;=\;1}^\infty$ , where

\begin{equation*}r_{nk}\;=\;\frac{p_k}{ P_n}, \ \text{for} \ k\le n,\end{equation*}

and zero otherwise.

For every sequence p such that

• $\frac{p_n}{P_n} \to 0,$ as $n\to\infty$ ,

• $\frac{1}{P_n} \sum\nolimits_{k\;=\;1}^n |p_k-p_{k+1}|\to 0,$ as $n\to\infty$ ,

the method of summation $R_p$ satisfies Corollary 3.6 by [Reference Boos3, Exercise 3.2.23] (these two bullets are exactly the strong regularity condition). For every sequence p satisfying two bullets above and such that p is monotonically increasing, the method of summation $R_p$ satisfies Corollary 3.5 by [Reference Boos3, Exercise 3.2.24].