Recent
Papers
These are papers about dense subsets of the integers, primes
or
integer lattices. They belong to the research area called
additive number theory/combinatorics.
Constellations in P^d (with
B.
Cook, Int. Math. Res. Notices, Advance Acess, 23pp, 06/2011
doi:10.1093/imrn/rnr127)
On restricted arithmetic progressions over
finite
fields (with B. Cook, submitted)
Optimal polynomial recurrence
(with N. Lyall, Can.J.Math., submitted)
Simultaneous polynomial recurrence
(with
N. Lyall, Bull. London Math. Soc.,v43/4, pp. 765-785, 2011)
k-point
configurations in sets of
positive density of Z^n (Duke Math. J., v 146/1, pp. 1-34,
2009)
Polynomial
configurations in difference sets (with N. Lyall, J.
Num. Theory, v. 129/2, pp.439-450, 2009)
On
distance
sets of large sets of integer
points (Israel J. Math., v 164/1, pp. 251-263, 2008)
These are two papers discussing discrete maximal functions, singular
Radon transforms and ergodic theorems related to Nilpotent groups.
(in joint work with A. Ionescu, E. M. Stein and S. 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:
On
Fourier
restriction and the Newton
polygon. (Proc. Amer. Math. Soc., v 137/2, pp. 615-625, 2009)
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.