Algebraic sums and products of univoque bases

Given x∈(0,1], let U(x) be the set of bases q∈(1,2] for which there exists a unique sequence (di) of zeros and ones such that x=∑i=1∞di∕qi. Lü et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x)+λU(x) and product U(x)⋅U(x)λ contain an interval for all x∈(0,1] and λ≠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 (Dajani and Kalle, 2017).