Additive combinatorics and incidence . . .

Additive Combinatorics is new discipline in mathematics with connections to additive number theory, fourier analysis, graph theory and probability. The field has numerous applications to various other fields, including Incidence Geometry (which focuses on the properties of lines and points in various geometries in a combinatorial sense). We consider the survey of Additive Combinatorics and its applications to Incidence Geometry by Zeev Dvir [1], and present in particular the Kakeya problem from Chapter 4 of [1]. The Kakeya problem deals with the rough notion of “size” of a subset of Rn or of Fn (or in general, any geometry with a well-defined notion of lines and direction) which has a “line” in every “direction”. We consider the cases of the reals and the finite fields. We examine the state of the art on the problem. This project was done in conjuction with Vijay Keswani (11799), a fourth year undergraduate in the Dept. of Computer Science and Engineering, who read on the Szemeredi-Trotter theorem on counting incidences between N points and N lines in a given geometry at the same time as when I was reading on the Kakeya problem, and both of us attended each others project presentations. The presentations were also attended by Prof. Shobha Madan from the Dept. of Mathematic