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arXiv · 2004.05926

Additive properties of numbers with restricted digits

Han Yu
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Abstract

In this paper, 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 following sense that for each $\epsilon>0,$ as $N\to\infty,$ \[ \#((B_{a}+B_{b})\cap B_{c} \cap [1,N])=O(N^{\epsilon}). \]

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Han Yu. 2020-04-13. Additive properties of numbers with restricted digits. https://doi.org/10.2140/ant.2021.15.1283

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