A remark on exceptional sets in Erdös-Rényi limit theorem

Abstract

Let $(x_i)_{i=1}^{+\infty }$ be the digits sequence in the unique terminating dyadic expansion of $x\in [0,1)$. The run-length function $l_n(x)$ is defined by

$$\begin{aligned} l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\ \text {for some}\ 0\le i\le n-j\right\} . \end{aligned}$$

Erdös and Rényi proved that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text {a.e.}\ x\in [0,1). \end{aligned}$$

In this note, we show that for each pair of numbers $\alpha ,\beta \in [0,+\infty ]$ with $\alpha \le \beta$, the following exceptional set

$$\begin{aligned} E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\ \limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \end{aligned}$$

has Hausdorff dimension one.