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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 |
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Vol. 17 (2023), No. 2, 497–539
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Abstract |
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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 . 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
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Mathematical Subject Classification
Primary: 11B30
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Milestones
Received: 8 October 2021
Revised: 21 February 2022
Accepted: 11 April 2022
Published: 24 March 2023
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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 |
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| Joni Teräväinen | |
| Mathematical Institute University of Oxford Radcliffe Observatory Quarter Oxford United Kingdom |
Department of Mathematics and
Statistics University of Turku Finland |
| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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