On convex equations

2023/10/14 by Tomasz Schoen, Schoen, Tomasz
Computer Science · Mathematics · #11B25 #11B30 #Advanced Mathematical Modeling in Engineering #Combinatorics (math.CO) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2310.09584

openalex publication_date 2023/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every subset of \1,…, 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.

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