New dimension bounds for $\alpha\beta$ sets

TL;DR

This paper establishes new lower bounds for the upper box dimension of $\

Contribution

It introduces novel bounds for the dimension of $\

Findings

If $\eta$ is a Liouville number and $\\\alpha$ is not, the dimension is 1.

New bounds improve understanding of the structure of $\\alpha\beta$ sets.

Results have implications for affine embeddings of self-similar sets.

Abstract

In this paper we obtain new lower bounds for the upper box dimension of $\alpha\beta$ sets. As a corollary of our main result, we show that if $\alpha$ is not a Liouville number and $\beta$ is a Liouville number, then the upper box dimension of any $\alpha\beta$ set is $1$. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.