  Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in
\{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of
non-negative integers, all of whose digits in base $b$ are either $d_1$ or
$d_2$. Then every sufficiently large integer is a sum of at most $b^{160 k^2}$
numbers of the form $x^k$, $x \in \mathcal{S}$.
