Proof of a conjecture of Kleinberg-Sawin-Speyer

IF 1 3区数学 Q1 MATHEMATICS

Discrete Analysis Pub Date : 2016-08-19 DOI:10.19086/DA.3733

Luke Pebody

{"title":"Proof of a conjecture of Kleinberg-Sawin-Speyer","authors":"Luke Pebody","doi":"10.19086/DA.3733","DOIUrl":null,"url":null,"abstract":"In Ellenberg and Gijswijt's groundbreaking work~\\cite{EllenbergGijswijt}, the authors show that a subset of $\\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\\epsilon)^n)$), and provide for any prime $p$ a value $\\lambda_p<p$ such that any subset of $\\mathbb{Z}_p^{n}$ with no arithmetic progression of length 3 must be of size at most $\\lambda_p^n$. \r\nBlasiak et al~\\cite{BlasiakEtAl} showed that the same bounds apply to tri-coloured sum-free sets, which are triples $\\{(a_i,b_i,c_i):a_i,b_i,c_i\\in\\mathbb{Z}_p^{n}\\}$ with $a_i+b_j+c_k=0$ if and only if $i=j=k$. \r\nBuilding on this work, Kleinberg, Sawin and Speyer~\\cite{KleinbergSawinSpeyer} gave a description of a value $\\mu_p$ such that no tri-coloured sum-free sets of size $e^{\\mu_p n}$ exist in $\\mathbb{Z}_p^{n}$, but for any $\\epsilon>0$, such sets of size $e^{(\\mu_p-\\epsilon) n}$ exist for all sufficiently large $n$. The value of $\\mu_p$ was left open, but a conjecture was stated which would imply that $e^{\\mu_p}=\\lambda_p$, i.e. the Ellenberg-Gijswijt bound is correct for the sum-free set problem. \r\nThe purpose of this note is to close that gap. The conjecture of Kleinberg, Sawin and Speyer is true, and the Ellenberg-Gijswijt bound is the correct exponent for the sum-free set problem.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2016-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"23","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.19086/DA.3733","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 23

Abstract

In Ellenberg and Gijswijt's groundbreaking work~\cite{EllenbergGijswijt}, the authors show that a subset of $\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\epsilon)^n)$), and provide for any prime $p$ a value $\lambda_p0$, such sets of size $e^{(\mu_p-\epsilon) n}$ exist for all sufficiently large $n$. The value of $\mu_p$ was left open, but a conjecture was stated which would imply that $e^{\mu_p}=\lambda_p$, i.e. the Ellenberg-Gijswijt bound is correct for the sum-free set problem. The purpose of this note is to close that gap. The conjecture of Kleinberg, Sawin and Speyer is true, and the Ellenberg-Gijswijt bound is the correct exponent for the sum-free set problem.

查看原文本刊更多论文

Kleinberg-Sawin-Speyer猜想的证明

在Ellenberg和Gijswijt的开创性工作\cite{EllenbergGijswijt}中，作者表明，长度为3的算式数列$\mathbb{Z}_3^{n}$的子集的大小必须最多为$2.755^n$(不知道$(3-\epsilon)^n)$的上界)，并且提供任何素数$p$ a值$\lambda_p0$，这样的大小集$e^{(\mu_p-\epsilon) n}$存在于所有足够大的$n$。$\mu_p$的值是开放的，但提出了一个猜想，这意味着$e^{\mu_p}=\lambda_p$，即Ellenberg-Gijswijt界对于无和集问题是正确的。本文的目的就是弥补这一差距。Kleinberg、Sawin和Speyer的猜想是正确的，Ellenberg-Gijswijt界是无和集问题的正确指数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Discrete Analysis Mathematics-Algebra and Number Theory

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

17 weeks

期刊介绍： Discrete Analysis is a mathematical journal that aims to publish articles that are analytical in flavour but that also have an impact on the study of discrete structures. The areas covered include (all or parts of) harmonic analysis, ergodic theory, topological dynamics, growth in groups, analytic number theory, additive combinatorics, combinatorial number theory, extremal and probabilistic combinatorics, combinatorial geometry, convexity, metric geometry, and theoretical computer science. As a rough guideline, we are looking for papers that are likely to be of genuine interest to the editors of the journal.