Digitally Restricted Sets and the Goldbach Conjecture: An Exceptional Set Result

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 $|\mathcal{A}(X)|^{1-\delta}$