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.The subject matter has advanced considerably since then, but the lectures may still be useful for learning some basic techniques.

Lecture #1: Kakeya problem, Kakeya maximal operator, and the restriction phenomenon: connections and relationships

Lecture #2: Adventures in the plane

Lecture #3: Higher dimensional adventures: n+1/2 and discrete n+2/2

Lecture #4: Wolff's n+2/2 result: We are in the 90's!

Lecture #5: Bourgain strikes again- Arithmetic Kakeya estimates
 

Fuglede conjecture for lattices:

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 paper was written around 2004.

Roth's theorem on arithmetic progressions:

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. This paper was written around 2002.

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. This paper was written around 2006.