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KAKEYA SETS OVER NON-ARCHIMEDEAN LOCAL RINGS

Part of: Finite geometry and special incidence structures Designs and configurations

Published online by Cambridge University Press:  28 March 2013

Evan P. Dummit  and
Márton Hablicsek
Evan P. Dummit
Affiliation:
UW-Madison, Department of Mathematics, 480 Lincoln Dr., Madison, WI 53706-1388,U.S.A. email dummit@math.wisc.edu
Márton Hablicsek
Affiliation:
UW-Madison, Department of Mathematics, 480 Lincoln Dr., Madison, WI 53706-1388,U.S.A. email hablics@math.wisc.edu

Abstract

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In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 1–25], the authors asked whether there are Besicovitch phenomena in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. Furthermore, we prove that any Kakeya set in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{2} $ or ${ \mathbb{Z} }_{p}^{2} $ is of Minkowski dimension 2.

MSC classification

Secondary: 05B30: Other designs, configurations 51E20: Combinatorial structures in finite projective spaces

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 257 - 266
Copyright
Copyright © University College London 2013 

References

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