{"title":"Furstenberg Sets in Finite Fields: Explaining and Improving the Ellenberg–Erman Proof","authors":"Manik Dhar, Zeev Dvir, Ben Lund","doi":"10.1007/s00454-023-00585-y","DOIUrl":null,"url":null,"abstract":"<p>A (<i>k</i>, <i>m</i>)-Furstenberg set is a subset <span>\\(S \\subset {\\mathbb {F}}_q^n\\)</span> with the property that each <i>k</i>-dimensional subspace of <span>\\({\\mathbb {F}}_q^n\\)</span> can be translated so that it intersects <i>S</i> in at least <i>m</i> points. Ellenberg and Erman (Algebra Number Theory <b>10</b>(7), 1415–1436 (2016)) proved that (<i>k</i>, <i>m</i>)-Furstenberg sets must have size at least <span>\\(C_{n,k}m^{n/k}\\)</span>, where <span>\\(C_{n,k}\\)</span> is a constant depending only <i>n</i> and <i>k</i>. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on <span>\\(C_{n,k}\\)</span>, and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension <span>\\(n-k\\)</span> varieties, instead of just co-dimension <span>\\(n-k\\)</span> subspaces.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"128 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-023-00585-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 3
Abstract
A (k, m)-Furstenberg set is a subset \(S \subset {\mathbb {F}}_q^n\) with the property that each k-dimensional subspace of \({\mathbb {F}}_q^n\) can be translated so that it intersects S in at least m points. Ellenberg and Erman (Algebra Number Theory 10(7), 1415–1436 (2016)) proved that (k, m)-Furstenberg sets must have size at least \(C_{n,k}m^{n/k}\), where \(C_{n,k}\) is a constant depending only n and k. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on \(C_{n,k}\), and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension \(n-k\) varieties, instead of just co-dimension \(n-k\) subspaces.
期刊介绍:
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.