Algebraic sums and products of univoque bases
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Publication date
2018-08
Authors
Dajani, K.
Kong, Derong
Komornik, Vilmos
Li, Wenxia
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Supervisors
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Article
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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).
Keywords
Algebraic differences, Non-integer base expansions, Univoque bases, Thickness, Cantor sets, Non-matching parameters, Taverne, General Mathematics
Citation
Dajani, K, Kong, D, Komornik, V & Li, W 2018, 'Algebraic sums and products of univoque bases', Indagationes Mathematicae, vol. 29, no. 4, pp. 1087–1104. https://doi.org/10.1016/j.indag.2018.05.010