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.
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.
Roth's theorem on
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
Falconer's distance set estimate via Stein-Tomas :
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.