## On the size of Kakeya sets in finite vector spaces.(English)Zbl 1295.11138

Summary: For a finite field $$\mathbb{F}_q$$, a Kakeya set $$K$$ is a subset of $$\mathbb{F}_q^n$$ that contains a line in every direction. This paper derives new upper bounds on the minimum size of Kakeya sets when $$q$$ is even.

### MSC:

 11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects) 11T06 Polynomials over finite fields

### Keywords:

Kakeya set; finite vector space; finite fields; value set
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### References:

 [1] A. W. Bluher. On xq+1+ ax + b. Finite Fields Appl., 10(3):285-305, 2004. · Zbl 1137.12300 [2] Z. Dvir. On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22(4):1093- 1097, 2009. · Zbl 1202.52021 [3] Z. Dvir. Incidence theorems and their applications. arXiv preprint arXiv:1208.5073, 2012. · Zbl 1278.68012 [4] Z. Dvir, S. Kopparty, S. Saraf, and M. Sudan. Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pages 181-190. IEEE Computer Soc., Los Alamitos, CA, 2009. · Zbl 1292.68119 [5] S. Kopparty, V. F. Lev, S. Saraf, and M. Sudan. Kakeya-type sets in finite vector spaces. J. Algebraic Combin., 34(3):337-355, 2011. · Zbl 1230.42027 [6] G. Kyureghyan and Q. Wang. An upper bound on the size of kakeya sets in finite vector spaces. to appear in proceedings of WCC 2013. · Zbl 1295.11138 [7] V.F.Lev.Amixingpropertyforfinitefieldsofcharacteristic2,2012.http://mathoverflow.net/questions/102751/ a-mixing-property-for-finite-fields-of-characteristic-2. the electronic journal of combinatorics 20(3) (2013), #P369 [8] R. J. McEliece. Finite fields for computer scientists and engineers. The Kluwer International Series in Engineering and Computer Science, 23. Kluwer Academic Publishers, Boston, MA, 1987. [9] G. Mockenhaupt and T. Tao. Restriction and Kakeya phenomena for finite fields. Duke Math. J., 121(1):35-74, 2004. · Zbl 1072.42007 [10] S. Saraf and M. Sudan. An improved lower bound on the size of Kakeya sets over finite fields. Anal. PDE, 1(3):375-379, 2008. · Zbl 1335.42017 [11] T. Wolff. Recent work connected with the Kakeya problem. In Prospects in mathematics (Princeton, NJ, 1996), pages 129-162. Amer. Math. Soc., Providence, RI, 1999. the electronic journal of combinatorics 20(3) (2013), #P3610 · Zbl 0934.42014
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