Mathematical Sciences Department Discrete Math Seminar
Speaker: Sam Adriaensen, Vrije Universiteit Brussel & WPI
Monday, February 24, 2025
9:00 - 9:50 am
Higgins Labs 202
Title: Sets determining few directions in finite planes
Abstract: Let F be a finite field of prime order p. Consider a set S of p points in the plane F2. We say that S determines direction (d) if there are two points in S which span a line with slope d. Here d is either an element of F or equals infinity in case of a vertical line X=b. The study of sets determining few directions was initiated by Rédei. S must determine at least 1 direction, and this happens exactly when S consists of all points on a line. Otherwise, a classical result states that S determines at least (p+3)/2 directions.
If |S| > p, then S determines every direction by the pigeonhole principle, but there is still a sensible way of linking directions to S. If |S| = k p for some k, we say that S is equidistributed from direction (d) if all lines with slope d intersect S in exactly k points. Otherwise, we call the direction special. Note that for k=1, a direction is special iff it is determined. Kiss and Somlai proved that for every prime p, there is a unique set of points (up to some equivalence) in F2 which has exactly 3 special directions.
This talk will be about joint work with Zsuzsa Weiner and Tamás Szőnyi, where we study the problem in a modular sense, and show the existence of multisets with exactly k special directions in the modular sense for every k > 2, which don't arise as unions of multisets with fewer special directions.