Algebraic sums and products of univoque bases

2017/10/09 by Karma Dajani, Dajani, Karma, Vilmos Komornik +5
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1710.03291

openalex publication_date 2017/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given x∈(0, 1], let \mathcal 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/qⁱ. 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)⋅\mathcal 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 (2017).

Citations

• Representations for real numbers and their ergodic properties
• Univoque bases and Hausdorff dimension

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