We prove a new bound on a version of the sum-product problem studied by Chang. By introducing several combinatorial tools, this expands upon a method of Croot and Hart which used the Tarry-Escott problem to build distinct sums from polynomials with specific vanishing properties. We also study other aspects of the sum-product problem such as a method to prove a dual to a result of Elekes and Ruzsa and a conjecture of J. Solymosi on combinatorial geometry. Lastly, we study two combinatorial problems on sumsets over the reals. The first involves finding Freiman isomorphisms of real-valued sets that also preserve the order of the original set. The second applies results from the former in proving a new Balog-Szemeredi type theorem for real-valued sets.
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/53950 |
| Date | 21 September 2015 |
| Creators | Bush, Albert |
| Contributors | Croot, Ernest |
| 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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