Abstract

Blocking Sets and Covers, variations of the Kakeya problem

Aart Blokhuis
Technische Universiteit Eindhoven

The finite field Kakeya problem asks for the (minimal) number of points covered by a set of lines in AG(n, q), the n-dimensional affine space over the field with q elements. It was essentially solved in a beautiful paper by Dvir, who proved the conjecture that the order of magnitude is q^n. The case n=2 was settled earlier. Here the precise lower bounds are known. For q even the lines can be taken `in general position' (folkore), for q odd one takes q lines in general position, and adds a suitable line in the last direction (with Mazzocca). Together with De Boeck, Mazzocca and Storme we determined the second smallest Kakeya set in the even case. A second variation we studied with Seva Lev. Fix n and k. We color the points of AG(n, 2), Red and Green, in such a way that for every point P there is a k-dimensional subspace containing it that is completely green, with the
possible exception of the point P itself. How many red points can there be? Equivalently, how many green points must there be. The dual problem of covering a set of points in the plane by lines, is to block a set of lines by points. Here the basic, but trivial result is that q + 1 points suffice to block all lines of the projective plane PG(2, q). For the afine plane the case is much more interesting and non-trivial. By a famous result of Jamison, Brouwer, Schrijver, we need at least 2q-1 points. With a relatively large group of people we recently looked at this problem for the set of points and blocks (secants) of the classical unital. This is a set of q3 + 1 points in PG(2, q^2), such that every line intersects it in q + 1 or 1 point.
It is well known that the points of PG(2, q^2) can be partitioned into q^2-q+1 Baer subplanes (PG(2, q)). The union of any t disjoint Baer subplanes forms a two-intersection set with intersection numbers t; q + t. With some luck I will also be able to say something interesting about the following problem: How many points are needed to cover the long secants (so the q + t-secants) of such a union.