Effective bounds for Roth's theorem with shifted square common difference

Sarah Peluse; Ashwin Sah; Mehtaab Sawhney

Effective bounds for Roth's theorem with shifted square common difference

Number Theory 2023-09-18 v1 Combinatorics

Authors: Sarah Peluse , Ashwin Sah , Mehtaab Sawhney

Abstract

Let $S$ be a subset of $\{1,\ldots,N\}$ avoiding the nontrivial progressions $x, x+y^2-1, x+ 2(y^2-1)$ . We prove that $|S|\ll N/\log_m{N}$ , where $\log_m$ is the $m$ -fold iterated logarithm and $m\in\mathbf{N}$ is an absolute constant. This answers a question of Green.

Keywords

multiplicative functions arithmetic progressions bound

Cite

@article{arxiv.2309.08359,
  title  = {Effective bounds for Roth's theorem with shifted square common difference},
  author = {Sarah Peluse and Ashwin Sah and Mehtaab Sawhney},
  journal= {arXiv preprint arXiv:2309.08359},
  year   = {2023}
}

Comments

47 pages

Effective bounds for Roth's theorem with shifted square common difference

Abstract

Keywords

Cite

Comments

Related papers