Roth--Waring--Goldbach

TL;DR

This paper applies Green's transference principle to demonstrate that dense subsets of prime powers contain solutions to certain linear equations, providing explicit bounds and extending understanding of additive properties of primes.

Contribution

It introduces a novel application of Green's transference principle to prime powers, establishing solutions to linear equations with explicit quantitative bounds.

Findings

Dense subsets of prime powers contain solutions to translation-invariant linear equations.

Explicit bounds are provided for the solutions.

The method extends additive number theory techniques to prime powers.

Abstract

We use Green's transference principle to show that any subset of the $d$th powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in $d^2+1$ or more variables, with explicit quantitative bounds.