Recent
Papers / Preprints
These are papers about subsets of the integer lattice of positive
upper density. They belong to the research area called density Ramsey
theory:
On
distance
sets of large sets of integer
points (Israel J. Math., v 164/1, 2008)
k-point
configurations in sets of
positive density of Z^n (to appear in Duke Math. J.)
Polynomial
configurations in difference sets (with N. Lyall, submitted to J.
Number Theory)
Optimal
polynomial return times (with N. Lyall,
preprint)
These are two papers discussing discrete maximal functions, singular
Radon transforms and ergodic theorems related to Nilpotent groups.
(in joint work with Alex Ionescu, Elias M. Stein and Steve Wainger):
Maximal
operators
associated to discrete subgroups of nilpotent Lie groups ( J.
d'Analyse Math., v. 101, pp. 257-313, 2007)
Discrete
Radon transforms and
applications to ergodic theory ( Acta Math., v. 198, No 2, pp.
231-298, 2007)
Here is a paper on the uniformity of distribution of integer points on
certain polynomial surfaces:
On
the distribution of lattice points on
spheres and on level surfaces of polynomials (J. Num. Theory,
v.122/1 pp. 69-83, 2007)
This preprint deals with the restriction of the Fourier transform to
two-dimensional analytic surfaces in R^3:
A
note on Fourier restriction and the Newton
polygon. (to appear in Proc. AMS.)
Here is a survey article which appeared in a book called "Fourier
analysis and Convexity" (ANHA, Birkahauser '04)
Discrete
maximal functions and ergodic
theorems related to polynomials.
The details of the proofs can be found in these two earlier papers:
"Diophantine
equations and ergodic theorems"
(Amer J. Math., v.124, p.921-953) and
"Discrete
analogues in harmonic analysis: spherical averages" (Annals
of
Math., v.155, p.189-208)
Here one can find some expository notes (jointly with Neil Lyall) and
some links on Ramsey
Theory.