  We determine the sets definable in expansions of the ordered real additive
group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a
set $C$ is a generalized Cantor set in base $r$ if there is a non-empty
$K\subseteq\{1,\ldots,r-2\}$ such that $C$ is the set of those numbers in
$[0,1]$ that admit a base $r$ expansion omitting the digits in $K$. While it is
known that the theory of an expansion of the ordered real additive group by a
single generalized Cantor set is decidable, we establish that the theory of an
expansion by two generalized Cantor sets in multiplicatively independent bases
is undecidable.
