Research

Interests

Analytic Number Theory; Harmonic Analysis; Automorphic Forms, Representations, and L-functions; 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 Local-Global 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: Low-Lying 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: Local-Global 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 Local-Global 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 L-functions, in Automorphic Representations and L-Functions, (Ed. by D. Prasad et al), Tata Institute, (2013), 263-310.
• (with Jean Bourgain) On Zaremba’s Conjecture, Annals Math 180 Issue 1, (2014), 137–196.
• (with Dorian Goldfeld and Eric Stade), On the Kontorovich-Lebedev Transform, Commentarii Mathematici 60, (2011), no. 1-2, (Fujii volume) 37-46.
• (with Jean Bourgain) On Zaremba’s Conjecture, research announcement, CR Math Paris 349 (2011) no. 9, 493-495.
• (with Dorian Goldfeld) On the Determination of the Plancherel Measure for Lebedev-Whittaker transforms on GL(n), Acta Arith 155 (Schinzel volume), 2012, pp. 15-26.
• (with Jean Bourgain) On a Theorem of Friedlander and Iwaniec, Comptes Rendus Math. Acad. Sci. Paris, Vol. 348, No. 17-18, Sept 2010, pp. 947-950.
• (with Jean Bourgain) On Representations of Integers in Thin Subgroups of SL(2,Z), GAFA Vol. 20, No.5, 2010, pp. 1144-1174.
  Erratum, GAFA Vol. 20, No.6, 2010, pp. 1548-1549.
• (with Jean Bourgain and Peter Sarnak) Sector Estimates for Hyperbolic Isometries, GAFA Vol. 20, No.5, 2010, pp. 1175-1200.
• (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 3-Manifolds, JAMS 24, (2011), no. 3, 603-648.
• (with Hee Oh) Almost Prime Pythagorean Triples in Thin Orbits, J. Reine Angew. Math., Vol 667 (2012), 89–131.
• (with Jeff Hoffstein) The First Non-Vanishing Quadratic Twist of an Automorphic L-series, preprint, 2009.
• The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves, Duke Math J., Vol. 149, No. 1, 2009, pp. 1-36.
• The Dirichlet Series for the Exterior Square L-function on GL(n), Ramanujan J. Vol 21, Issue 3 (2010).
• A Pseudo-Twin Primes Theorem, in “Multiple Dirichlet Series, L-functions and Automorphic Forms”, Birkhauser Progress in Math Series, Vol. 300, (2012), 287–298.
• (with Aviezri Fraenkel) The Sierpinski Sieve of Nim-varieties 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. 1-13.
• (with Steven J. Miller) Benford’s Law, values of L-functions and the 3x+1 Problem, Acta Arithmetica 120 (2005), 269-297.
• (with Yakov Sinai) Structure Theorem for (d,g,h)-Maps, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.