A transference principle for systems of linear equations, and applications to almost twin primes

TL;DR

This paper develops a general transference principle for affine-linear configurations of finite complexity, enabling the transfer of combinatorial results to sparse prime sets, with applications to almost twin primes and primes of specific forms.

Contribution

It introduces a broad transference principle applicable to affine-linear configurations, extending prior results to new prime sets like Chen primes and primes of the form x^2 + y^2 + 1.

Findings

Existence of solutions in prime sets depends on local conditions.

Application to almost twin primes and primes of specific algebraic forms.

Relies on recent nilsequence estimates for prime distributions.

Abstract

The transference principle of Green and Tao enabled various authors to transfer Szemer\'edi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of primes the existence of solutions to finite complexity systems of linear equations is determined by natural local conditions. These applications rely on a recent work of the last two authors on Bombieri-Vinogradov type estimates for nilsequences.