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KAKEYA SETS OVER NON-ARCHIMEDEAN LOCAL RINGS

  • Evan P. Dummit (a1) and Márton Hablicsek (a2)

Abstract

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.

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References

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1.Besicovitch, A., Sur deux questions d’intégrabilité des fonctions. J. Soc. Phys. Math. 2 (1919), 105123.
2.Davies, R. O., Some remarks on the Kakeya problem. Proc. Cambridge Phil. Soc. 69 (1971), 417421.
3.Dvir, Z., On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22 (2009), 10931097.
4.Ellenberg, J.  S., Oberlin, R. and Tao, T., The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 125.
5.Wolff, T., An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoam. 11 (1999), 651674.
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KAKEYA SETS OVER NON-ARCHIMEDEAN LOCAL RINGS

  • Evan P. Dummit (a1) and Márton Hablicsek (a2)

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