Vol. 15, No. 5, 2021

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Additive properties of numbers with restricted digits

Han Yu

Vol. 15 (2021), No. 5, 1283–1301
Abstract

We consider some additive properties of integers with restricted digit expansions. Let b 3 be an integer and Bb 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 (Ba + Bb) Bc. In particular, under mild conditions, (Ba + Bb) Bc is a very thin set in the sense that for each 𝜖 > 0, as N ,

#((Ba + Bb) Bc [1,N]) = O(N𝜖).

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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