Vinogradov three prime 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 $1\le i\le 3$. 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 $c_i$.

42 pages