On rich and poor directions determined by a subset of a finite plane

Abstract

We generalize to sets with cardinality more than p a theorem of Rédei and Szőnyi on the number of directions determined by a subset U of the finite plane F 2 p . A U -rich line is a line that meets U in at least #U/p + 1 points, while a U -poor line is one that meets U in at most #U/p − 1 points. The slopes of the U -rich and U -poor lines are called U -special directions. We show that either U is contained in the union of n = ⌈#U/p⌉ lines, or it determines "many" Uspecial directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the Rédei polynomial to take into account the "multiplicity" of the directions determined by U .

Special directions on the finite affine plane

“…In this section we will try to find sets of smallest possible cardinality, having exactly 4 special directions. For small prime p ď 11 we construct such sets of minimal cardinality according to Ghidelli's lower bound [4].…”

“…What is the minimal size of a set in F 2 p , which is equidistributed in at most k directions? In particular, is it true that Ghidelli's bound is tight [4], i.e., 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 2 0 1 1 1 1 1 1 1 1 0 1 1 2 0 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 is it possible to construct sets of cardinality kp which have r p`k`2 k`1 s special directions?…”

“…The investigation of sets of cardinality larger than p was initiated by Ghidelli [4]. It was proved that a set of cardinality kp (1 ď k ď p, k P Z) is either a set of parallel lines or is not equidistributed in at least r p`k`2 k`1 s directions.…”