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arXiv · 1905.00832

Digit expansions of numbers in different bases

Stuart A. Burrell·Han Yu
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Abstract

A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in $[0, 1]$ who do not contain some digit in their $b$-expansion for all $b \geq 3$ has zero Hausdorff dimension.

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Stuart A. Burrell, Han Yu. 2019-05-02. Digit expansions of numbers in different bases. https://doi.org/10.1016/j.jnt.2021.01.003

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