Sets of rich lines in general position

Abstract

AbstractThe Szemerédi–Trotter theorem implies that the number of lines incident to at least of points in is . J. Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position—no two parallel, no three meeting at a point—then one can get a much tighter bound. We prove a slight variant of his conjecture: for every there exists some such that for sufficiently large values of , every set of lines in general position, each intersecting an grid of points in at least places, has size at most . This implies a conjecture of Gy. Elekes about the existence of a uniform statistical version of Freiman's theorem for linear functions with small image sets.