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 ( d i ) of zeros and ones such that x = ∑ i = 1 ∞ d i ∕ 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 ) + λ U ( 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). [ABSTRACT FROM AUTHOR]