Georgia Institute of Technology College of Sciences

We demonstrate new results in additive combinatorics, including a proof of the following conjecture by J. Solymosi: for every epsilon > 0, there exists delta > 0 such that, given n^2 points in a grid formation in R^2, 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.