TY - JOUR
T1 - A transference principle for systems of linear equations, and applications to almost twin primes
AU - Bienvenu, Pierre Yves
AU - Shao, Xuancheng
AU - Teräväinen, Joni
N1 - Publisher Copyright:
© 2023 MSP (Mathematical Sciences Publishers).
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - higher order Fourier analysis
KW - Szemerédi’s theorem
UR - http://www.scopus.com/inward/record.url?scp=85152276296&partnerID=8YFLogxK
U2 - 10.2140/ant.2023.17.497
DO - 10.2140/ant.2023.17.497
M3 - Article
AN - SCOPUS:85152276296
SN - 1937-0652
VL - 17
SP - 497
EP - 539
JO - Algebra and Number Theory
JF - Algebra and Number Theory
IS - 2
ER -