## 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 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.

Original language | English |
---|---|

Pages (from-to) | 497-539 |

Number of pages | 43 |

Journal | Algebra and Number Theory |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

### Bibliographical note

Funding Information:Bienvenu is grateful for the financial support and hospitality of the Max Planck Institute for Mathematics, Bonn. While finishing up he was supported by the joint FWF-ANR project Arithrand: FWF: I 4945-N and ANR-20-CE91-0006. Shao was supported by the NSF grant DMS-1802224. Teräväinen was supported by a Titchmarsh Research Fellowship. We thank the anonymous referee for a thorough reading of the paper and many insightful comments and suggestions.

Publisher Copyright:

© 2023 MSP (Mathematical Sciences Publishers).

## Keywords

- Szemerédi’s theorem
- higher order Fourier analysis

## ASJC Scopus subject areas

- Algebra and Number Theory