Update on 2-dimensional Kakeya sets over finite fields

A few months ago I wrote about the problem of giving a sharp lower bound for the size of a Kakeya subset of F_q^2; that is, a subset containing a line in every direction. Apparently this problem has now been solved: Simeon Ball, in a comment on Terry’s blog, says that Blokhuis and Mazzocca show that a Kakeya set has have cardinality at least q(q+1)/2 + (q-1)/2. Since there are known examples of Kakeya sets of this size arising from conics, this bound is sharp. The argument appears to be completely combinatorial, not algebro-geometric a la Dvir.

I’m a bit confused about attribution; Ball calls it a result of Blokhuis and Mazzocca, but links to a preprint of his own which apparently proves the theorem, and which doesn’t mention Blokhuis and Mazzocca. So who actually proved this nice result? Anyone with first-hand knowledge should enlighten me in comments.

Tagged , , , , , ,

3 thoughts on “Update on 2-dimensional Kakeya sets over finite fields

  1. The results by A.Blokhuis and F.Mazzocca about the finite Kakeya problem in two dimensional case, cited by Simeon Ball, will appear on the last volume of Bolyai Society Mathematical Studies dedicated to L.Lovász’s 60th birthday ( http://www.springer.com/math/numbers/book/978-3-540-85218-6?detailsPage=toc )

  2. none says:

    Dvir has a new survey article online: http://www.eccc.uni-trier.de/report/2009/077/

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 576 other followers

%d bloggers like this: