  In this paper, we consider some additive properties of integers with
restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set
of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$
be three integers greater than $2.$ We give some estimates on the size of
$(B_{a}+B_{b})\cap B_{c}.$ In particular, under mild conditions,
$(B_{a}+B_{b})\cap B_{c}$ is a very thin set in the following sense that for
each $\epsilon>0,$ as $N\to\infty,$
  \[
  \#((B_{a}+B_{b})\cap B_{c} \cap [1,N])=O(N^{\epsilon}).
  \]
