Arithmetic Progressions with Restricted Digits

TL;DR

This paper investigates the presence of arithmetic progressions within Kempner sets, which are sparse digit-restricted sets of integers, and determines the maximum length of such progressions that exclude a specific digit.

Contribution

It precisely characterizes the maximum length of arithmetic progressions in Kempner sets that omit a given digit for all bases.

Findings

Exact maximum length of progressions omitting a digit in Kempner sets

Characterization valid for all bases

Advances understanding of additive structures in digit-restricted sets

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.