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arXiv · 1710.03291

Algebraic sums and products of univoque bases

Karma Dajani·Vilmos Komornik·Derong Kong·Wenxia Li
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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 $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. 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).

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Karma Dajani, Vilmos Komornik, Derong Kong, Wenxia Li. 2018-07-11. Algebraic sums and products of univoque bases. https://arxiv.org/abs/1710.03291

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