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.
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/53090 |
| Date | 12 January 2015 |
| Creators | Pryby, Christopher Ian |
| Contributors | Croot, Ernie |
| Publisher | Georgia Institute of Technology |
| Source Sets | Georgia Tech Electronic Thesis and Dissertation Archive |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
| Format | application/pdf |
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