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

Part of: Harmonic analysis in several variables Arithmetic algebraic geometry Finite geometry and special incidence structures

Published online by Cambridge University Press:  10 December 2009

Jordan S. Ellenberg ,
Richard Oberlin  and
Terence Tao
Jordan S. Ellenberg
Affiliation:
Department of Mathematics, University of Wisconsin, Madison WI 53706, U.S.A. (email: ellenber@math.wisc.edu)
Richard Oberlin
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A. (email: oberlin@math.ucla.edu)
Terence Tao
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A. (email: tao@math.ucla.edu)
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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.

MSC classification

Secondary: 42B25: Maximal functions, Littlewood-Paley theory 11G25: Varieties over finite and local fields 51E20: Combinatorial structures in finite projective spaces
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 1 - 25
Copyright
Copyright © University College London 2010

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