Algebraic sums and products of univoque bases

Abstract

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 /q i . 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).