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A transference principle for systems of linear equations, and applications to almost twin primes
- Source :
- Alg. Number Th. 17 (2023) 231-273
- Publication Year :
- 2021
-
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.<br />Comment: 37 pages, to appear in Algebra and Number Theory
- Subjects :
- Mathematics - Number Theory
11B30, 11N36
Subjects
Details
- Database :
- arXiv
- Journal :
- Alg. Number Th. 17 (2023) 231-273
- Publication Type :
- Report
- Accession number :
- edsarx.2106.09001
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2140/ant.2023.17.231