Algebraic sums and products of univoque bases

TL;DR

This paper investigates the algebraic sums and products of univoque bases, showing they contain intervals, which reveals new structural properties of these sets related to unique expansions.

Contribution

It demonstrates that algebraic sums and products of univoque bases contain intervals, extending understanding of their algebraic and topological structure.

Findings

Algebraic sum $rac{rac{rac{U(x)+rac{rac{rac{U(x)}{rac{rac{rac{U(x)}{rac{rac{rac{U(x)}{rac{rac{rac{U(x)+rac{rac{rac{U(x)}}{rac{rac{rac{U(x)}} contains an interval.

The product $rac{rac{rac{U(x)}{rac{rac{rac{U(x)}^rac{rac{rac{U(x)}}$ also contains an interval.

This phenomenon applies to the set of non-matching parameters studied in prior work.

Abstract

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+\lambda\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^\lambda$ contain an interval for all $x\in(0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017).