Year: 2023
Type: preprint
Abstract: We prove that every subset of $\{1,\dots, N\}$ which does not contain any solutions to the equation $x+y+z=3w$ has at most $\exp(-c(\log N)^{1/5+o(1)})N$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally, our method has the potential to yield an optimal estimate for this problem that matches the known Behrend's lower estimate. Our approach relies on a new result on almost-periodicity of convolutions.
Source: arXiv (Cornell University)
Author: Schoen, Tomasz
Institution: Adam Mickiewicz University in Poznań
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387723747