Algebraic sums and products of univoque bases

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 $x={\sum }_{i=1}^{\infty }{d}_i/q^i$. Lü, 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).

21 pages, 1 figure. To appear in Indag. Math