Additive stucture, rich lines, and exponential set-expansion

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.

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Georgia Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Borenstein, Evan

Advisor dc:contributor.advisor

• Croot, Ernest

Committee members dc:contributor.committeemember

• Costello, Kevin
• Lyall, Neil
• Tetali, Prasad
• Yu, XingXing

• Arithmetic combinatorics
• Additive combinatorics
• Combinatorics
• Incidence geometry
• Sum-product inequalities
• Structural theorems

Handle dc:identifier.uri

http://hdl.handle.net/1853/29664

OAI identifier oai:identifier

oai:repository.gatech.edu:1853/29664