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Sets of rich lines in general position
Author(s) -
Amirkhanyan Gagik,
Bush Albert,
Croot Ernest,
Pryby Chris
Publication year - 2017
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12051
Subject(s) - general position , conjecture , combinatorics , position (finance) , mathematics , grid , upper and lower bounds , point (geometry) , set (abstract data type) , discrete mathematics , geometry , mathematical analysis , computer science , finance , economics , programming language
The Szemerédi–Trotter theorem implies that the number of lines incident to at least k > 1 of n points in R 2 is O ( n 2 / k 3 + n / k ) . 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 ε > 0 there exists some δ > 0 such that for sufficiently large values of n , every set of lines in general position, each intersecting an n × n grid of points in at least n 1 − δplaces, has size at most n ε . 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.