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Research interests

Most of my research is in Ergodic Theory and Dynamical Systems, and Number Theory.

My focus in dynamics is on group actions on homogeneous spaces. For example, I am interested in rigidity phenomena for actions by "large" groups, like higher-rank abelian groups or semisimple Lie groups.

Homogeneous dynamics interacts strongly with number theory, especially Diophantine approximation. Some of my current projects are in this area of interaction, as well as other related areas, like fractal geometry.

Papers

Khintchine's Theorem with random fractions,
Preprint.

A characterization of bad approximability,
Submitted.

Rational approximation of affine coordinate subspaces of Euclidean space,
with D. S. Simmons and F. Sueess,
Acta Arithmetica 177 (2017), no. 1, 91-100.

Different speeds suffice for rendezvous of two agents in arbitrary graphs,
with E. Kranakis, D. Krizanc, E. Markou, and A. Pagourtzis,
SOFSEM 2017.

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation,
International Journal of Number Theory 13 (2017), no. 3, 633-654.

Metric Diophantine Approximation: aspects of recent work,
with V. Beresnevich and S. Velani,
Dynamics and Analytic Number Theory, London Mathematical Society Lecture Note Series 437, eds. D. Badziahin, A. Gorodnik, N. Peyerimhoff, Cambridge University Press, 2016. (pp. 1-95)

Khintchine types of translated coordinate hyperplanes,
Acta Arithmetica 170 (2015), no. 3, 243-273.

Recurrence to shrinking targets on typical self-affine fractals,
with H. Koivusalo,
Proceedings of the Edinburgh Mathematical Society, to appear.

Higher cohomology of parabolic actions on certain homogeneous spaces,
Journal d'Analyse Mathématique 131 (2017), no. 1, 255-275.

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions,
Journal of Functional Analysis 265 (2013), no. 6, 1002-1063.

Limit theorems for rank-one Lie groups,
with A. Gorodnik,
Proceedings of the AMS 142 (2014), no. 4, 1359-1369.

Higher cohomology for Anosov actions on certain homogeneous spaces,
Journal d'Analyse Mathématique 121 (2013), no. 1, 177-222.

Smooth cocycles over homogeneous dynamical systems,
Ph. D. Thesis (2010).

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups,
Journal of Modern Dynamics 3 (2009), no. 3, 335-357.