On the range of simple symmetric random walks on the line

This paper is aimed at a detailed study of the behaviors of random walks which is defined by the dyadic expansions of points. More precisely, let $x=({ϵ}_1\left(x\right),{ϵ}_2\left(x\right),\dots )$ be the dyadic expansion for a point $x\in 0,1)$ and $S_n\left(x\right)={\sum }_{k=1}^n(2{ϵ}_k\left(x\right)-1)$, which can be regarded as a simple symmetric random walk on $\mathbb{Z}.$ Denote by $R_n\left(x\right)$ the cardinality of the set $\{S_1\left(x\right),\dots ,S_n\left(x\right)\},$ which is just the distinct position of $x$ passed after $n$ times. The set of points whose behavior satisfies $R_n\left(x\right)\sim c{n}^{\gamma }$ is studied ($c>0$ and $0<\gamma \le 1$ being fixed) and its Hausdorff dimension is calculated.