Vinogradov’s Theorem with Piatetski-Shapiro Primes

Abstract We prove that, for any $c_{1},c_{2},c_{3}\in (1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes, $N = p_{1} + p_{2} +p_{3}$, such that $p_{i} = \lfloor n_{i}^{c_{i}}\rfloor $ for some $n_{i} \in \mathbb N$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green’s transference principle due to Matomäki, Maynard, and Shao. We prove a necessary restriction estimate using Bourgain’s strategy and employ Harman’s sieve method to optimize our upper bound for $c_{i}$.