Arithmetic Progressions with Restricted Digits. Issue 2 (7th February 2020) - Walker, Aled; Walker, Alexander
Abstract: For an integer b ⩾ 2 and a set S ⊂ 0, …, b − 1 , we define the Kempner set K ( S, b ) 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.