  We show that for any set $D$ of at least two digits in a given base $b$,
there exists a $\delta(D,b)>0$ such that within the set $\mathcal{A}$ of
numbers whose digits base $b$ are exclusively from $D$, the number of even
integers in $\mathcal{A}$ which are less than $X$ and not representable as the
sum of two primes is less than $|\mathcal{A}(X)|^{1-\delta}$
