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The Finite Field Kakeya Problem

Surprisingly there exist figures with arbitrarily small area that fulfill Kakeya's conditions. This was discovered soon after A. S. Besicovitch constructed a set, containing a line segment in every direction, of zero Lesbesgue measure. Such a construction can be found in [1] or [5]. It was subsequently conjectured that a Kakeya set in RWn (a compact set in RWn containing a line segment in every direction) has fractal dimension n. In the plane, this has been proved in the affirmative and a proof can be found in [5]. For higher dimensions however, the problem is still open, and only in the last seven years has much progress been made. Recent work of Bourgain [2] and Katz and Tao [3] revealed a lower bound of 4n/7 + 3/7 on the Minkowski dimension of Kakeya sets in R n. This, when n is large, is an improvement on Wolff's bound of n/2 + 1 in [4]. The conjecture has important links to harmonic analysis and number theory; see [5] for a survey. It plays a significant role in the theory of oscillatory integrals and is of relevance to questions related to the distribution of Dirichlet series. In [5] Wolff introduced finite field Kakeya sets and posed the corresponding Kakeya conjecture. This reduction removes much of the technicality (there is no use of measure or fractal dimension), while preserving the essence of the problem. We apply the ideas of Bourgain, Katz, and Tao to the finite field version of the problem. Let IF be a field with'q elements. A Kakeya set in Fn is a set E C En containing a line in every direction; thus for all e E I'\{0} there is an a E Fn such that a + te E E for all t E= F. It seems reasonable to conjecture that finite field Kakeya sets have bounded cardinality: