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1	Additive stucture, rich lines, and exponential set-expansion Borenstein, Evan 19 May 2009 (has links) We will survey some of the major directions of research in arithmetic combinatorics and their connections to other fields. We will then discuss three new results. The first result will generalize a structural theorem from Balog and Szemerédi. The second result will establish a new tool in incidence geometry, which should prove useful in attacking combinatorial estimates. The third result evolved from the famous sum-product problem, by providing a partial categorization of bivariate polynomial set functions which induce exponential expansion on all finite sets of real numbers. Arithmetic combinatorics Additive combinatorics Combinatorics Incidence geometry Sum-product inequalities Structural theorems Combinatorial analysis Combinatorial geometry Set functions