Recent Papers

Discrete multilinear maximal operators and pinned simplices (with N. Lyall, A. Newman, P. Woolfitt, submitted)

Polynomial averages and pointwise ergodic theorems on nilpotent groups, I: Step two groups (with A. Ionescu. M.A. Mirek, T. Szarek, submitted)

Simplices in thin subsets of Eucliden spaces (with A. Iosevich, Anal. PDE, to appear)

Weak hypergraph regularity and applications to geometric Ramsey theory (with N. Lyall, Trans. Amer. Math. Soc., to appear)

The Discrete Spherical Maximal Function A new proof of L^2-boundedness (with N. Lyall, A. Newman, P. Woolfitt, Proc. AMS, v. 149/12, pp. 5305-5312, 2021)

Discrete multilinear maximal operators associated to simplices (with B. Cook, N. Lyall, J. London Math. Soc., v 104/4, pp.1491-1514, 2021)

Distances and trees in dense subsets of Z^d  (with N. Lyall, Israel J. Math, pp.1-22, 2020)

Distance graphs and sets of positive upper density in R^d (with N. Lyall, Anal. PDE v. 13/3 pp. 385-400, 2020)

Spherical configurations over finite fields (with N. Lyall, H. Parshall, Amer. J. Math. v. 142/2. pp. 373-404, 2020 )

A multidimensional Szemeredi theorem in the primes (with B. Cook, T. Titichetrakun, Annals of Combinatorics 22/4, pp.711-768, 2018)

Product of simplices and sets of positive density in R^d (with N. Lyall, Math. Proc. Cambridge Phil. Soc., v 165/1, pp. 25-51, 2018

A Roth type theorem for dense subsets of R^d (with B. Cook, M. Pramanik, Bull. London Math. Soc., v 49/4, pp. 1473-1489, 2017 )

Embedding simplices into sets of positive upper density in R^d (with L. Huckaba,  N. Lyall, Proc. A.M.S., 145, pp. 2335-2447, 2017)

Almost prime solutions of diophantine systems of high rank (with T. Titichetrakun, Int. J. Num. Th., v 13/5, pp. 1491-1514, 2017)

Diophantine equations in the primes (with B. Cook, Invent. Math.: v 198/3, pp. 701-737, 2014)

Averages along polynomial sequences in discrete nilpotent groups: Singular Radon transforms
(with A. Ionescu, S. Wainger, in Advances in Analysis: The Legacy of E. M. Stein, Princeton Univ. Press, pp. 147-188, 2014)

On the distribution of solutions to diophantine equations (in A Panorama of Discrepancy Theory, Springer, pp. 461-505, 2014)

Optimal polynomial recurrence (with N. Lyall, Canad. J. Math. 65, 171-194, 2013)

Constellations in P^d (with B. Cook, Int. Math. Res. Notices, v 12 pp. 2794-2816, 2012)

On restricted arithmetic progressions over finite fields (with B. Cook, OJM, v 7, pp. 1-10, 2012)

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.