THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS

Abstract

MathematikaVolume 56, Issue 1 p. 1-25 Research Article THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS Jordan S. Ellenberg, Jordan S. Ellenberg [email protected] Department of Mathematics, University of Wisconsin, Madison, WI, 53706 U.S.A.Search for more papers by this authorRichard Oberlin, Richard Oberlin [email protected] UCLA Department of Mathematics, Los Angeles, CA, 90095-1555 U.S.A.Search for more papers by this authorTerence Tao, Terence Tao [email protected] UCLA Department of Mathematics, Los Angeles, CA, 90095-1555 U.S.A.Search for more papers by this author Jordan S. Ellenberg, Jordan S. Ellenberg [email protected] Department of Mathematics, University of Wisconsin, Madison, WI, 53706 U.S.A.Search for more papers by this authorRichard Oberlin, Richard Oberlin [email protected] UCLA Department of Mathematics, Los Angeles, CA, 90095-1555 U.S.A.Search for more papers by this authorTerence Tao, Terence Tao [email protected] UCLA Department of Mathematics, Los Angeles, CA, 90095-1555 U.S.A.Search for more papers by this author First published: 10 December 2009 https://doi.org/10.1112/S0025579309000400Citations: 20AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Using the polynomial method of Dvir [On the size of Kakeya sets in finite fields. Preprint], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties W over finite fields F. For instance, given an (n−1)-dimensional projective variety W⊂¶n(F), we establish the Kakeya maximal estimate for all functions f:Fn→R and d≥1, where for each w∈W, the supremum is over all irreducible algebraic curves in Fn of degree at most d that pass through w but do not lie in W, and with Cn,W,d depending only on n,d and the degree of W; the special case when W is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir. Citing Literature Volume56, Issue1January 2010Pages 1-25 This article also appears in:Fields Medal Collection RelatedInformation

Locations

• Mathematika - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• DataCite API - View
• Mathematika - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• DataCite API - View
• Mathematika - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• DataCite API - View