Additive stucture, rich lines, and exponential set-expansion

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

Borenstein, Evan ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Arithmetic combinatorics;    Additive combinatorics;    Combinatorics;    Incidence geometry;    Sum-product inequalities;    Structural theorems;   

Others  :  https://smartech.gatech.edu/bitstream/1853/29664/1/borenstein_evan_s_200908_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

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.