  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.
