Vinogradov three prime theorem with Piatetski-Shapiro primes

TL;DR

This paper proves that sufficiently large odd numbers can be expressed as sums of three Piatetski-Shapiro primes within certain exponents, extending the classical Vinogradov theorem to a new prime subset.

Contribution

It introduces a novel approach combining Green's transference principle, Bourgain's restriction estimates, and Harman's sieve to handle Piatetski-Shapiro primes in Vinogradov's theorem.

Findings

Established representation of large odd numbers as sums of three Piatetski-Shapiro primes.

Extended Vinogradov's theorem to primes of the form loor(n^{c_i}) for specific c_i.

Developed new bounds and methods for primes with fractional power constraints.

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 Matom\"aki, 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 $c_i$.