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Arithmetic Progressions with Restricted Digits

Abstract For an integer and a set , we define the Kempner set to be the set of all nonnegative 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 article 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.