Research
Interests
Analytic Number Theory; Harmonic Analysis; Automorphic Forms, Representations, and Lfunctions; Hyperbolic Geometry and Spectral Theory; Group Actions and Ergodic Theory; Sieve and Circle Methods
Papers

 (with Kei Nakamura) Geometry and Arithmetic of Crystallographic Sphere Packings, submitted, 2017.
 The LocalGlobal Principle for Integral Soddy Sphere Packings, submitted, 2017.
 (with Arthur Baragar) Efficiently constructing tangent circles, submitted, 2017.
 (with Jean Bourgain) Beyond Expansion IV: Traces of Thin SemiGroups, submitted, 2017.
 (with Jean Bourgain) Beyond Expansion III: Reciprocal Geodesics, submitted, 2016.
 Applications of Thin Orbits, submitted, 2016.
 (with Jean Bourgain and Michael Magee) Thermodynamic expansion to arbitrary moduli, to appear, Crelle, 2017.
 (with Dubi Kelmer) Effective equidistribution of shears and applications, to appear, Math Annalen, 2017.
 (with Peter McNamara and Geordie Williamson) Appendix to Schubert calculus and torsion explosion by Geordie Williamson, to appear, JAMS, 2016.
 (with Jiuzu Hong) Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits, Israel J. Math, 209 no. 1, (2015) 397–420.
 (with Jean Bourgain) Beyond Expansion II: LowLying Fundamental Geodesics, JEMS, 19 (2017), 1331–1359.
 Levels of Distribution and the Affine Sieve, Ann. Fac. Sci. Toulouse Math 23, (2014), no. 5, 933–966.
 (with Dubi Kelmer) On the Pair Correlation Density for Hyperbolic Angles, Duke Math J, (2015), no. 3, 473–509.
 (with Jean Bourgain) The Affine Sieve Beyond Expansion I: Thin Hypotenuses, IMRN (19), 9175–9205 (2015).
 (with Valentin Blomer, Farrell Brumley, and Nicolas Templier) Bounding hyperbolic and spherical coefficients of Maass forms, J. Theor. Nombres Bordeaux 26, (2014), no. 3, 559–579.
 From Apollonius To Zaremba: LocalGlobal Phenomena in Thin Orbits, Bull. AMS 50, (2013), No. 2, 187–228.
 The Orbital Circle Method, in “Thin Groups and Super Approximation”, MSRI Publications Vol 61, Cambridge University Press (2014), 93–106.
 (with Jean Bourgain) On the LocalGlobal Conjecture for Integral Apollonian Gaskets, Inventiones Math, 196 (2014), no. 3, 589–650.
 (with Dorian Goldfeld) On the GL(3) Kuznetsov formula with applications to symmetry types of families of Lfunctions, in Automorphic Representations and LFunctions, (Ed. by D. Prasad et al), Tata Institute, (2013), 263310.
 (with Jean Bourgain) On Zaremba’s Conjecture, Annals Math 180 Issue 1, (2014), 137–196.
 (with Dorian Goldfeld and Eric Stade), On the KontorovichLebedev Transform, Commentarii Mathematici 60, (2011), no. 12, (Fujii volume) 3746.
 (with Jean Bourgain) On Zaremba’s Conjecture, research announcement, CR Math Paris 349 (2011) no. 9, 493495.
 (with Dorian Goldfeld) On the Determination of the Plancherel Measure for LebedevWhittaker transforms on GL(n), Acta Arith 155 (Schinzel volume), 2012, pp. 1526.
 (with Jean Bourgain) On a Theorem of Friedlander and Iwaniec, Comptes Rendus Math. Acad. Sci. Paris, Vol. 348, No. 1718, Sept 2010, pp. 947950.

 (with Jean Bourgain) On Representations of Integers in Thin Subgroups of SL(2,Z), GAFA Vol. 20, No.5, 2010, pp. 11441174.
Erratum, GAFA Vol. 20, No.6, 2010, pp. 15481549.
 (with Jean Bourgain) On Representations of Integers in Thin Subgroups of SL(2,Z), GAFA Vol. 20, No.5, 2010, pp. 11441174.

 (with Jean Bourgain and Peter Sarnak) Sector Estimates for Hyperbolic Isometries, GAFA Vol. 20, No.5, 2010, pp. 11751200.
 (with Jeff Lagarias) Stochastic Models for the 3x + 1 and 5x + 1 Problems, in “The Ultimate Challenge: The 3x+1 Problem,” AMS 2010.
 (with Hee Oh) Apollonian Circle Packings and Closed Horospheres on Hyperbolic 3Manifolds, JAMS 24, (2011), no. 3, 603648.

 (with Hee Oh) Almost Prime Pythagorean Triples in Thin Orbits, J. Reine Angew. Math., Vol 667 (2012), 89–131.

 (with Jeff Hoffstein) The First NonVanishing Quadratic Twist of an Automorphic Lseries, preprint, 2009.

 The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves, Duke Math J., Vol. 149, No. 1, 2009, pp. 136.

 The Dirichlet Series for the Exterior Square Lfunction on GL(n), Ramanujan J. Vol 21, Issue 3 (2010).

 A PseudoTwin Primes Theorem, in “Multiple Dirichlet Series, Lfunctions and Automorphic Forms”, Birkhauser Progress in Math Series, Vol. 300, (2012), 287–298.

 (with Aviezri Fraenkel) The Sierpinski Sieve of Nimvarieties and Binomial Coefficients, Combinatorial number theory, 209–227, de Gruyter, Berlin, 2007.

 (with Mel Nathanson) Quadratic addition rules for quantum integers, Journal of Number Theory, Vol 117, Issue 1, March 2006, pp. 113.

 (with Steven J. Miller) Benford’s Law, values of Lfunctions and the 3x+1 Problem, Acta Arithmetica 120 (2005), 269297.
 (with Yakov Sinai) Structure Theorem for (d,g,h)Maps, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213224.