Expository papers

Kakeya Lectures:

A series of lectures on the Kakeya problem, the restriction conjecture, and related issues, delivered at the probability learning seminar at the University of Missouri-Columbia during the first semester of 2000.

This short note gives a simple proof of a theorem due to Fuglede which says that the statement that the lattice L tiles a domain in Euclidean space is equivalent to the statement that the dual lattice generates an orthogonal basis of exponentials for L_2 of this domain. An even shorter proof can be given using the Poisson Summation Formula. The proof here was designed to be almost completely self-contained, so the PSF is essentially reproved in the course of the argument.

This note contains a short proof of the celebrated Szemeredi-Trotter incidences theorem which says that the number of incidences between n points and m lines does not exceed a constant multiple of  n+m+(nm)^{2/3}. Several applications are given to problems in harmonic analysis, convex geometry and additive number theory. Several distance sets results are proved using geometric methods. Connection with the Falconer Distance Conjecture is provided and used to deduce discrete distance set results using Fourier methods.

In this note we give a simple and self-contained proof of Roth's theorem which says that the any subset of the positive integers of positive density contains an arithmetic progression of length three.

Kakeya in finite fields (notes written for the University of Missouri program for high school students in August 2004). Note that these notes are hopelessly outdated since the Kakeya conjecture in vector spaces over finite fields has been solved by Zeev Dvir.

In this note we use the Stein-Tomas restriction theorem to prove a result due to Falconer which says that if the Hausdorff dimension of a set is greater than (d+1)/2, then the Lebesgue measure of the set of distances is positive. This point of view has let to some interesting advances in recent years in papers by Bourgain, Wolff, Erdogan and others.

Erdos distance problem (notes writte for the University of Missouri program for high school students in August 2005).