{"title":"关于有界度关系的限制集合","authors":"Minghui Ouyang","doi":"10.1112/mtk.70045","DOIUrl":null,"url":null,"abstract":"<p>Given two subsets <span></span><math></math> and a binary relation <span></span><math></math>, the restricted sumset of <span></span><math></math> with respect to <span></span><math></math> is defined as <span></span><math></math>. When <span></span><math></math> is taken as the equality relation, determining the minimum value of <span></span><math></math> is the famous Erdős–Heilbronn problem, which was solved separately by Dias da Silva, Hamidoune and Alon, Nathanson and Ruzsa. Lev later conjectured that if <span></span><math></math> with <span></span><math></math> and <span></span><math></math> is a matching between subsets of <span></span><math></math> and <span></span><math></math>, then <span></span><math></math>. We confirm this conjecture in the case where <span></span><math></math> for any <span></span><math></math>, provided that <span></span><math></math> for some sufficiently large <span></span><math></math> depending only on <span></span><math></math>. Our proof builds on a recent work by Bollobás, Leader, and Tiba, and a rectifiability argument developed by Green and Ruzsa. Furthermore, our method extends to cases when <span></span><math></math> is a degree-bounded relation, either on both sides <span></span><math></math> and <span></span><math></math> or solely on the smaller set. In addition, we construct subsets <span></span><math></math> with <span></span><math></math> such that <span></span><math></math> for any prime number <span></span><math></math>, where <span></span><math></math> is a matching on <span></span><math></math>. This extends an earlier construction by Lev and highlights a distinction between the combinatorial notion of the restricted sumset and the classcial Erdős–Heilbronn problem, where <span></span><math></math> holds given <span></span><math></math> is the equality relation on <span></span><math></math> and <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On restricted sumsets with bounded degree relations\",\"authors\":\"Minghui Ouyang\",\"doi\":\"10.1112/mtk.70045\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given two subsets <span></span><math></math> and a binary relation <span></span><math></math>, the restricted sumset of <span></span><math></math> with respect to <span></span><math></math> is defined as <span></span><math></math>. When <span></span><math></math> is taken as the equality relation, determining the minimum value of <span></span><math></math> is the famous Erdős–Heilbronn problem, which was solved separately by Dias da Silva, Hamidoune and Alon, Nathanson and Ruzsa. Lev later conjectured that if <span></span><math></math> with <span></span><math></math> and <span></span><math></math> is a matching between subsets of <span></span><math></math> and <span></span><math></math>, then <span></span><math></math>. We confirm this conjecture in the case where <span></span><math></math> for any <span></span><math></math>, provided that <span></span><math></math> for some sufficiently large <span></span><math></math> depending only on <span></span><math></math>. Our proof builds on a recent work by Bollobás, Leader, and Tiba, and a rectifiability argument developed by Green and Ruzsa. Furthermore, our method extends to cases when <span></span><math></math> is a degree-bounded relation, either on both sides <span></span><math></math> and <span></span><math></math> or solely on the smaller set. In addition, we construct subsets <span></span><math></math> with <span></span><math></math> such that <span></span><math></math> for any prime number <span></span><math></math>, where <span></span><math></math> is a matching on <span></span><math></math>. 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引用次数: 0
摘要
给定两个子集和一个二元关系,关于的限制和集定义为。当取为等式关系时,确定的最小值就是著名的Erdős-Heilbronn问题,Dias da Silva、Hamidoune and Alon、Nathanson and Ruzsa分别解决了这个问题。Lev后来推测,如果与和是与的子集之间的匹配,则。我们在任何情况下证实了这个猜想,假设对于一些足够大的只依赖于。我们的证明基于Bollobás、Leader和Tiba最近的一项工作,以及Green和Ruzsa提出的可纠错性论证。此外,我们的方法扩展到当是一个度有界的关系时,要么在两边,要么只在较小的集合上。此外,我们构造了这样的子集:对于任何素数,其中有一个匹配。这扩展了Lev早期的构造,并突出了限制集合的组合概念与经典Erdős-Heilbronn问题之间的区别,其中给定的是和上的相等关系。
On restricted sumsets with bounded degree relations
Given two subsets and a binary relation , the restricted sumset of with respect to is defined as . When is taken as the equality relation, determining the minimum value of is the famous Erdős–Heilbronn problem, which was solved separately by Dias da Silva, Hamidoune and Alon, Nathanson and Ruzsa. Lev later conjectured that if with and is a matching between subsets of and , then . We confirm this conjecture in the case where for any , provided that for some sufficiently large depending only on . Our proof builds on a recent work by Bollobás, Leader, and Tiba, and a rectifiability argument developed by Green and Ruzsa. Furthermore, our method extends to cases when is a degree-bounded relation, either on both sides and or solely on the smaller set. In addition, we construct subsets with such that for any prime number , where is a matching on . This extends an earlier construction by Lev and highlights a distinction between the combinatorial notion of the restricted sumset and the classcial Erdős–Heilbronn problem, where holds given is the equality relation on and .
期刊介绍:
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.