{"id":{"repo_id":"gatech","oai_identifier":"oai:repository.gatech.edu:1853/29664"},"canonical_url":"https://search.dev.ndltd.org/etd/gatech/oai:repository.gatech.edu:1853/29664","repository":{"repo_id":"gatech","name":"Georgia Tech","base_url":"https://repository.gatech.edu/server/oai/request"},"display":{"title":"Additive stucture, rich lines, and exponential set-expansion","abstract":"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. 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