在有限仿射空间 AG(n,2) 中避免给定大小的交集

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2024-09-24 DOI:10.1016/j.jcta.2024.105959

{"title":"在有限仿射空间 AG(n,2) 中避免给定大小的交集","authors":"","doi":"10.1016/j.jcta.2024.105959","DOIUrl":null,"url":null,"abstract":"<div><div>We study the set of intersection sizes of a <em>k</em>-dimensional affine subspace and a point set of size <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></math></span> of the <em>n</em>-dimensional binary affine space <span><math><mrow><mi>AG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in <em>k</em>-dimensional affine subspaces are unavoidable in all <em>m</em>-element point sets in the <em>n</em>-dimensional affine space.</div><div>We also show constructions of point sets for which the intersection sizes with <em>k</em>-dimensional affine subspaces take values from a set of a small size compared to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.</div><div><em>Keywords</em>: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000980/pdfft?md5=62687b67d599290d3f204041642a9a6a&pid=1-s2.0-S0097316524000980-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Avoiding intersections of given size in finite affine spaces AG(n,2)\",\"authors\":\"\",\"doi\":\"10.1016/j.jcta.2024.105959\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We study the set of intersection sizes of a <em>k</em>-dimensional affine subspace and a point set of size <span><math><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></math></span> of the <em>n</em>-dimensional binary affine space <span><math><mrow><mi>AG</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in <em>k</em>-dimensional affine subspaces are unavoidable in all <em>m</em>-element point sets in the <em>n</em>-dimensional affine space.</div><div>We also show constructions of point sets for which the intersection sizes with <em>k</em>-dimensional affine subspaces take values from a set of a small size compared to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.</div><div><em>Keywords</em>: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.</div></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-09-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000980/pdfft?md5=62687b67d599290d3f204041642a9a6a&pid=1-s2.0-S0097316524000980-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000980\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000980","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们研究了 k 维仿射子空间与 n 维二元仿射空间 AG(n,2) 大小为 m∈[0,2n] 的点集的交集大小集。按照厄尔多斯、富雷迪、罗斯柴尔德和 T. 索斯的主题，我们部分确定了 k 维仿射子空间中的哪些局部密度在 n 维仿射空间的所有 m 元素点集中是不可避免的。这些都是由仿射子空间和所谓的子空间规避集建立起来的。同时，我们改进了关于子空间逃避集的已知上界，并应用了关于数字的规范带符号数字（CSD）表示的结果。关键词：不可避免、仿射子空间、逃避集、随机方法、规范带符号数字系统。

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

Avoiding intersections of given size in finite affine spaces AG(n,2)

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size

m \in [0, 2^{n}]

of the n-dimensional binary affine space

AG (n, 2)

. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all m-element point sets in the n-dimensional affine space.

We also show constructions of point sets for which the intersection sizes with k-dimensional affine subspaces take values from a set of a small size compared to

2^{k}

. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.

Keywords: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.

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

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.