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PreprintCC BY 4.0arXiv

$2$-large sets are sets of nil-Bohr recurrence

  • Ryan Alweiss
  • Sohail Farhangi
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Abstract

A set S of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in S. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood B of $0$ and every ε>0, restricting the common difference to B increases the extremal number for k-term arithmetic progressions by at most a factor of 1+ε, provided k is sufficiently large.

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Cite

BibTeXRIS
BibTeX
@misc{alweiss20262large,
  title = {{$2$-large sets are sets of nil-Bohr recurrence}},
  author = {Ryan Alweiss and Sohail Farhangi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16605},
  eprint = {2610.16605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16605},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - $2$-large sets are sets of nil-Bohr recurrence
AU  - Ryan Alweiss
AU  - Sohail Farhangi
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.16605
UR  - https://arxiv.org/abs/2610.16605
AB  - A set S of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in S. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood B of $0$ and every ε>0, restricting the common difference to B increases the extremal number for k-term arithmetic progressions by at most a factor of 1+ε, provided k is sufficiently large.
ER  -