On arithmetic progressions in self-similar sets

Given a sequence $\{b_i{\}}_{i=1}^n$ and a ratio $\lambda \in (0,1),$ let $E={\cup }_{i=1}^n(\lambda E+b_i)$ be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in $E$. Our main idea is from the multiple $\beta$-expansions.