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arXiv · 2009.13795

Irrational self-similar sets

Qi Jia·Yuanyuan Li·Kan Jiang
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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.

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Qi Jia, Yuanyuan Li, Kan Jiang. 2020-09-29. Irrational self-similar sets. https://arxiv.org/abs/2009.13795

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