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

摘要

We generalize to sets with cardinality more than p a theorem of Redei and Szonyi on the number of directions determined by a subset U of the finite plane F-p(2). 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 = left perpendicular#U /pright perpendicular lines, or it determines "many" U -special directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the Redei polynomial to take into account the "multiplicity" of the directions determined by U. (C) 2020 Elsevier B.V. All rights reserved.