Some results on sums and products

Additive combinatorics;Incidence geometry;Sum-product inequality

Pryby, Christopher Ian ; Croot, Ernie Mathematics Lacey, Michael Lyall, Neil Yu, Xingxing Trotter, William T. ; Croot, Ernie

University:Georgia Institute of Technology

Department:Mathematics

关键词: Additive combinatorics;    Incidence geometry;    Sum-product inequality;   

Others  :  https://smartech.gatech.edu/bitstream/1853/53090/1/PRYBY-DISSERTATION-2014.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We demonstrate new results in additive combinatorics, including a proof of a conjecture by J. Solymosi: for every epsilon > 0, there exists delta > 0 such that, given n² points in a grid formation in R², if L is a set of lines in general position such that each line intersects at least n^{1-delta} points of the grid, then |L| < n^epsilon. This result implies a conjecture of Gy. Elekes regarding a uniform statistical version of Freiman's theorem for linear functions with small image sets.