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arXiv · 1809.02430

Arithmetic Progressions with Restricted Digits

Aled Walker·Alexander Walker
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Abstract

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.

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Aled Walker, Alexander Walker. 2018-09-07. Arithmetic Progressions with Restricted Digits. https://arxiv.org/abs/1809.02430

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