Arithmetic Progressions with Restricted Digits.

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. [ABSTRACT FROM AUTHOR]