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

Pierre-Yves Bienvenu, Xuancheng Shao and Joni Teräväinen

Vol. 17 (2023), No. 2, 497–539
Abstract

The transference principle of Green and Tao enabled various authors to transfer Szemerédi’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 x2 + y2 + 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.

Keywords
Szemerédi's theorem, higher order Fourier analysis
Mathematical Subject Classification
Primary: 11B30
Milestones
Received: 8 October 2021
Revised: 21 February 2022
Accepted: 11 April 2022
Published: 24 March 2023
Authors
Pierre-Yves Bienvenu
Institut für Analysis und Zahlentheorie
TU Graz
Kopernikusgasse 24/II
8010 Graz
Austria
School of Mathematics
Trinity College Dublin
Dublin 2
Ireland
Xuancheng Shao
Department of Mathematics
University of Kentucky
Lexington, KY
United States
Joni Teräväinen
Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Oxford
United Kingdom
Department of Mathematics and Statistics
University of Turku
Finland

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