Irrational self-similar sets

TL;DR

This paper investigates self-similar sets with zero Lebesgue measure, showing that for almost every translation, the shifted set contains only irrational or transcendental numbers, and constructs specific examples using q-expansions.

Contribution

It explicitly constructs translations of certain self-similar sets that contain only irrational or transcendental numbers, expanding understanding of their arithmetic properties.

Findings

For almost every translation, the shifted set contains only irrationals or transcendental numbers.

Explicit constructions of such translations are provided for specific classes of self-similar sets.

The approach uses q-expansions to analyze the properties of the shifted sets.

Abstract

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or transcendental numbers. In this note, we shall consider some classes of self-similar sets, and explicitly construct such $t$'s. Our main idea is from the $q$-expansions.