# On arithmetic progressions in self-similar sets

## TL;DR

This paper investigates the presence and maximum length of arithmetic progressions within self-similar sets generated by specific affine transformations, using techniques from multiple beta-expansions.

## Contribution

It introduces a novel approach connecting arithmetic progressions in self-similar sets with multiple beta-expansions, providing new insights into their structure.

## Key findings

- Characterization of arithmetic progressions in self-similar sets
- Bounds on the maximal length of such progressions
- Application of multiple beta-expansions to fractal analysis

## Abstract

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.