This bound is tight as seen by the conic construction. Let C be a conic in PG(2,q), L the line at infinity, and m a bisecant to C (a line is incident with two points of the conic). Let E be the external points to C (which are the points that are incident with two tangents of C). Then the set
(E U C U m) \ L is a Kakeya (Besikovitch) set with q(q+1)/2+(q-1)/2 points.

Moreover, Blokhuis and Mazzocca have shown that all such sets that attain the bound come from this construction. The bound follows from an old result of Jamison. I have detailed this in
http://www-ma4.upc.es/~simeon/kakeya.pdf
To classify the sets meeting the bound, you use similar arguments to those used by Segre in the fifties to prove that a set of q+1 points in PG(2,q), q odd, with the property that every line is incident with at most two points of the set, is a conic. The proof of this has been done by Blokhuis and Mazzocca, although I’m not sure this preprint is available yet.

Dear Cliff,

In two dimensions, there are reasonably sharp results (giving sets close to q^2/2) in

http://arxiv.org/abs/math/0510356
http://arxiv.org/abs/math/0607734

(and see also a comment in

http://quomodocumque.wordpress.com/2008/03/25/give-the-people-who-like-the-kakeya-problem-what-they-want/

)

In higher dimensions, the best construction I know of uses the two-dimensional construction, combined with the trivial observation that the Cartesian product of two Kakeya sets is still a Kakeya set (in a higher dimensional space, of course). This gives about q^d / 2^{d/2} in the case when d is even, and slightly worse than this when d is odd. This is a fair distance away from Dvir’s bound, which is about q^d/d!. So there is still a little bit of room to close the gap here, and in particular to settle the question of whether the density of Kakeya sets needs to decay faster than exponentially (i.e. it is asymptotically less than c^d q^d as d -> infinity for any c > 0).

What are some upper bounds on the minimum size of a Kakeya set in F_q^n?

I found one of size (q-1)^n + 2^n - 1. Surely there are smaller ones?

Cliff

One nice problem for practicing discretization+polynomial method or other methods is this: What is the largest measure of a subset of the -sphere which does not contain two orthogonal vectors. a natural guess is the twice the measure of a spherical cup of radius . When is large this is roughly a fraction of the total measure of the sphere.

Now, the corresponding problem for the discrete cube (when the dimension is 4 times a prime number) can be solved (up to a factor 2 or so) by the polynomial method based on calculations over the field of 2 elements. This result translates to a result for the Euclidean question which is the best known but it is far from the conjecture. (It gives
measure . ) This is essentially Frankl-Wilson’s theorem.

One can try to discretize differently and use the polynomial methods for other fields and finding an appropriate “weighting” is indeed part of the difficulty but so far nobody managed to do it. There are promising competing methods which at present gives weaker bounds.

From past experience with HJ I would have to say that how one defines the “number” of lines involved is going to be very important; a naive notion of cardinality is probably not the right one (it assigns too much weight to those lines with about half of the coordinates being active). A randomly formulated problem in this area is likely to be either trivially true, trivially false, or otherwise not terribly interesting.

One can certainly interpret this proof in coding language; the proposition in the post can be viewed as an assertion that a Reed-Solomon code of a certain order can be uniquely decoded by looking at the restriction of that code to a Kakeya set, and the Lemma can be viewed as a lower bound on the capacity of that code. I believe that this proof of Kakeya was motivated by some computer science considerations that are somewhat related to coding theory, but I do not know the full details.

Dear Gil: Well, there is of course the density Hales-Jewett theorem of Furstenberg and Katznelson, which (viewed contrapositively) is the assertion that if a set contains a point in every combinatorial line, then it has asymptotically full measure, which is an assertion of a vaguely Kakeya-type nature. Of course, algebraic methods such as the polynomial method are unlikely to apply directly to such a combinatorial situation that has no obvious algebraic structure, but it is still an interesting question as to whether some other proof of the density Hales-Jewett theorem (including, perhaps, a simplified ergodic theory proof) is available.