Vinogradov's Theorem for Primes With Restricted Digits

Let $g$ be sufficiently large, $b\in \{0,\ldots ,g-1\}$, and $\mathcal{S}_{b}$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_{1} + p_{2} + p_{3}$ where $p_{i}$ are prime and $p_{i}\in \mathcal{S}_{b}$.