Sumset大小和结构的有效结果

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2023-09-18 DOI:10.1007/s00493-023-00055-2

Andrew Granville, George Shakan, Aled Walker

{"title":"Sumset大小和结构的有效结果","authors":"Andrew Granville, George Shakan, Aled Walker","doi":"10.1007/s00493-023-00055-2","DOIUrl":null,"url":null,"abstract":"Let \\(A \\subset {\\mathbb {Z}}^d\\) be a finite set. It is known that NA has a particular size (\\(\\vert NA\\vert = P_A(N)\\) for some \\(P_A(X) \\in {\\mathbb {Q}}[X]\\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when \\(d=1\\), when the convex hull of A is a simplex or when \\(\\vert A\\vert = d+2\\) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"13 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Effective Results on the Size and Structure of Sumsets\",\"authors\":\"Andrew Granville, George Shakan, Aled Walker\",\"doi\":\"10.1007/s00493-023-00055-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let \\\\(A \\\\subset {\\\\mathbb {Z}}^d\\\\) be a finite set. It is known that NA has a particular size (\\\\(\\\\vert NA\\\\vert = P_A(N)\\\\) for some \\\\(P_A(X) \\\\in {\\\\mathbb {Q}}[X]\\\\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when \\\\(d=1\\\\), when the convex hull of A is a simplex or when \\\\(\\\\vert A\\\\vert = d+2\\\\) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.\",\"PeriodicalId\":50666,\"journal\":{\"name\":\"Combinatorica\",\"volume\":\"13 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00493-023-00055-2\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-023-00055-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 3

摘要

设\（A\subet｛\mathbb｛Z｝｝^d\）是一个有限集。已知，一旦N大于某个阈值，对于某些（P_a（X）\ in｛\mathbb｛Q｝｝[X]\）和结构（锥中除某些例外集之外的所有格点），NA具有特定的大小（\（\vert NA\vert=P_a（N）\）。在本文中，我们给出了任意A的该阈值的第一个有效上界。这种显式结果以前只有在特殊情况下才知道，当\（d=1\），当A的凸包是单纯形，或者当\（\vert A\vert=d+2\）Curran和Goldmaher（Discrete Anal.Paper No.272021），我们改进了结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Effective Results on the Size and Structure of Sumsets

Let \(A \subset {\mathbb {Z}}^d\) be a finite set. It is known that NA has a particular size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in {\mathbb {Q}}[X]\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when \(d=1\), when the convex hull of A is a simplex or when \(\vert A\vert = d+2\) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.