Vinogradov's Theorem for Primes With Restricted Digits.

The article establishes a generalization of Vinogradov’s three primes theorem by proving that every sufficiently large odd integer \( N \) can be expressed as the sum of three primes whose base-\( g \) expansions omit a fixed digit \( b \), provided the base \( g \) is sufficiently large. The main result quantifies the asymptotic count of such representations using Fourier analytic methods combined with zero-density estimates for Dirichlet \( L \)-functions and a refined Fourier approximation of the von Mangoldt function. Key technical contributions include combinatorial lemmas ensuring uniformity in digit restrictions, large sieve inequalities for measures supported on restricted digit sets, and the handling of correction terms arising from zeros of \( L \)-functions. The work relies on decomposing digit patterns into product measures with favorable Fourier decay and applies advanced character sum estimates to control error terms, culminating in an explicit formula for the number of representations with primes avoiding the digit \( b \) in base \( g \).