Additive properties of numbers with restricted digits

Yu, Han

doi:10.2140/ant.2021.15.1283

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Additive properties of numbers with restricted digits

Han Yu

Vol. 15 (2021), No. 5, 1283–1301

DOI: 10.2140/ant.2021.15.1283

Abstract

We consider some additive properties of integers with restricted digit expansions. Let $b \geq 3$ be an integer and $B_{b}$ be the set of integers whose base $b$ expansions have only digits ${0, 1}$ . Let $a, b, c$ be three integers greater than $2$ . We give some estimates on the size of $(B_{a} + B_{b}) \cap B_{c}$ . In particular, under mild conditions, $(B_{a} + B_{b}) \cap B_{c}$ is a very thin set in the sense that for each $𝜖 > 0$ , as $N \to \infty$ ,

# ((B_{a} + B_{b}) \cap B_{c} \cap [1, N]) = O (N^{𝜖}) .

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Keywords

dynamical system, Diophantine equation, Furstenberg's intersection problem

Mathematical Subject Classification

Primary: 11A63, 11K55, 28A80, 28D05, 37C45

Milestones

Received: 15 April 2020

Revised: 13 November 2020

Accepted: 12 December 2020

Published: 30 June 2021

Authors

Han Yu
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom

Vol. 15, No. 5, 2021