{"id":365,"date":"2020-07-09T16:56:57","date_gmt":"2020-07-09T16:56:57","guid":{"rendered":"https:\/\/sites.rutgers.edu\/demo-professor\/?page_id=365"},"modified":"2020-07-09T19:10:58","modified_gmt":"2020-07-09T19:10:58","slug":"research","status":"publish","type":"page","link":"https:\/\/sites.rutgers.edu\/demo-professor\/research\/","title":{"rendered":"Research"},"content":{"rendered":"<h2>Interests<\/h2>\n<p>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<\/p>\n<h2>Papers<\/h2>\n<ul>\n<li>(with Kei Nakamura)\u00a0<a href=\"https:\/\/arxiv.org\/abs\/1712.00147\" target=\"_blank\" rel=\"noopener noreferrer\">Geometry and Arithmetic of Crystallographic Sphere Packings<\/a>, submitted, 2017.<\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/1208.5441\" target=\"_blank\" rel=\"noopener noreferrer\">The Local-Global Principle for Integral Soddy Sphere Packings<\/a>, submitted, 2017.<\/li>\n<li>(with Arthur Baragar)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1704.08747\" target=\"_blank\" rel=\"noopener noreferrer\">Efficiently constructing tangent circles<\/a>, submitted, 2017.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1310.7190\" target=\"_blank\" rel=\"noopener noreferrer\">Beyond Expansion IV: Traces of Thin SemiGroups<\/a>, submitted, 2017.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1610.07260\" target=\"_blank\" rel=\"noopener noreferrer\">Beyond Expansion III: Reciprocal Geodesics<\/a>, submitted, 2016.<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1606.06325\" target=\"_blank\" rel=\"noopener noreferrer\">Applications of Thin Orbits<\/a>, submitted, 2016.<\/li>\n<li>(with Jean Bourgain and Michael Magee)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1507.07993\" target=\"_blank\" rel=\"noopener noreferrer\">Thermodynamic expansion to arbitrary moduli<\/a>, to appear, Crelle, 2017.<\/li>\n<li>(with Dubi Kelmer)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1506.05534\" target=\"_blank\" rel=\"noopener noreferrer\">Effective equidistribution of shears and applications<\/a>, to appear, Math Annalen, 2017.<\/li>\n<li>(with Peter McNamara and Geordie Williamson) Appendix to\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1309.5055\" target=\"_blank\" rel=\"noopener noreferrer\">Schubert calculus and torsion explosion<\/a>\u00a0by Geordie Williamson, to appear, JAMS, 2016.<\/li>\n<li>(with Jiuzu Hong)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1401.4701\" target=\"_blank\" rel=\"noopener noreferrer\">Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits<\/a>, Israel J. Math, 209 no. 1, (2015) 397&#8211;420.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1406.1366\" target=\"_blank\" rel=\"noopener noreferrer\">Beyond Expansion II: Low-Lying Fundamental Geodesics<\/a>, JEMS, 19 (2017), 1331&#8211;1359.<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/1406.1375\" target=\"_blank\" rel=\"noopener noreferrer\">Levels of Distribution and the Affine Sieve<\/a>, Ann. Fac. Sci. Toulouse Math 23, (2014), no. 5, 933&#8211;966.<\/li>\n<li>(with Dubi Kelmer)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1308.0754\" target=\"_blank\" rel=\"noopener noreferrer\">On the Pair Correlation Density for Hyperbolic Angles<\/a>, Duke Math J, (2015), no. 3, 473&#8211;509.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1307.3535\" target=\"_blank\" rel=\"noopener noreferrer\">The Affine Sieve Beyond Expansion I: Thin Hypotenuses<\/a>, IMRN (19), 9175&#8211;9205 (2015).<\/li>\n<li>(with Valentin Blomer, Farrell Brumley, and Nicolas Templier)\u00a0<a href=\"http:\/\/www.math.univ-paris13.fr\/~brumley\/Research_files\/BBKT%20--%20May%2031,%202013.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Bounding hyperbolic and spherical coefficients of Maass forms<\/a>, J. Theor. Nombres Bordeaux 26, (2014), no. 3, 559&#8211;579.<\/li>\n<li><a href=\"http:\/\/sites.math.rutgers.edu\/~alexk\/files\/Kontorovich2012.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits<\/a>, Bull. AMS 50, (2013), No. 2, 187&#8211;228.<\/li>\n<li><a href=\"http:\/\/library.msri.org\/books\/Book61\/files\/30kont.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">The Orbital Circle Method<\/a>, in &#8220;Thin Groups and Super Approximation&#8221;, MSRI Publications Vol 61, Cambridge University Press (2014), 93&#8211;106.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1205.4416\" target=\"_blank\" rel=\"noopener noreferrer\">On the Local-Global Conjecture for Integral Apollonian Gaskets<\/a>, Inventiones Math, 196 (2014), no. 3, 589&#8211;650.<\/li>\n<li>(with Dorian Goldfeld)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1203.6667\" target=\"_blank\" rel=\"noopener noreferrer\">On the GL(3) Kuznetsov formula with applications to symmetry types of families of L-functions<\/a>, in Automorphic Representations and L-Functions, (Ed. by D. Prasad et al), Tata Institute, (2013), 263-310.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1107.3776\" target=\"_blank\" rel=\"noopener noreferrer\">On Zaremba&#8217;s Conjecture<\/a>, Annals Math 180 Issue 1, (2014), 137&#8211;196.<\/li>\n<li>(with Dorian Goldfeld and Eric Stade), On the Kontorovich-Lebedev Transform, Commentarii Mathematici 60, (2011), no. 1-2, (Fujii volume) 37-46.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1103.0422\" target=\"_blank\" rel=\"noopener noreferrer\">On Zaremba&#8217;s Conjecture<\/a>, research announcement, CR Math Paris 349 (2011) no. 9, 493-495.<\/li>\n<li>(with Dorian Goldfeld)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1102.5086\" target=\"_blank\" rel=\"noopener noreferrer\">On the Determination of the Plancherel Measure for Lebedev-Whittaker transforms on GL(n)<\/a>, Acta Arith 155 (Schinzel volume), 2012, pp. 15-26.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1008.0825\" target=\"_blank\" rel=\"noopener noreferrer\">On a Theorem of Friedlander and Iwaniec<\/a>, Comptes Rendus Math. Acad. Sci. Paris, Vol. 348, No. 17-18, Sept 2010, pp. 947-950.<\/li>\n<li>(with Jean Bourgain)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1001.4534\" target=\"_blank\" rel=\"noopener noreferrer\">On Representations of Integers in Thin Subgroups of SL(2,Z)<\/a>, GAFA Vol. 20, No.5, 2010, pp. 1144-1174.<br \/>\n<a href=\"http:\/\/sites.math.rutgers.edu\/~alexk\/files\/BK2010Erratum.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Erratum<\/a>, GAFA Vol. 20, No.6, 2010, pp. 1548-1549.<\/li>\n<li>(with Jean Bourgain and Peter Sarnak)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1001.4541\" target=\"_blank\" rel=\"noopener noreferrer\">Sector Estimates for Hyperbolic Isometries<\/a>, GAFA Vol. 20, No.5, 2010, pp. 1175-1200.<\/li>\n<li>(with Jeff Lagarias)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/0910.1944\" target=\"_blank\" rel=\"noopener noreferrer\">Stochastic Models for the 3x + 1 and 5x + 1 Problems<\/a>, in &#8220;The Ultimate Challenge: The 3x+1 Problem,&#8221; AMS 2010.<\/li>\n<li>(with Hee Oh)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/0811.2236\" target=\"_blank\" rel=\"noopener noreferrer\">Apollonian Circle Packings and Closed Horospheres on Hyperbolic 3-Manifolds<\/a>, JAMS 24, (2011), no. 3, 603-648.<\/li>\n<li>(with Hee Oh)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1001.0370\" target=\"_blank\" rel=\"noopener noreferrer\">Almost Prime Pythagorean Triples in Thin Orbits<\/a>, J. Reine Angew. Math., Vol 667 (2012), 89&#8211;131.<\/li>\n<li>(with Jeff Hoffstein)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1008.0839\" target=\"_blank\" rel=\"noopener noreferrer\">The First Non-Vanishing Quadratic Twist of an Automorphic L-series<\/a>, preprint, 2009.<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/0712.1391\" target=\"_blank\" rel=\"noopener noreferrer\">The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves<\/a>, Duke Math J., Vol. 149, No. 1, 2009, pp. 1-36.<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/0909.4869\" target=\"_blank\" rel=\"noopener noreferrer\">The Dirichlet Series for the Exterior Square L-function on GL(n)<\/a>, Ramanujan J. Vol 21, Issue 3 (2010).<\/li>\n<li><a href=\"http:\/\/arxiv.org\/abs\/math\/0507569\" target=\"_blank\" rel=\"noopener noreferrer\">A Pseudo-Twin Primes Theorem<\/a>, in &#8220;Multiple Dirichlet Series, L-functions and Automorphic Forms&#8221;, Birkhauser Progress in Math Series, Vol. 300, (2012), 287&#8211;298.<\/li>\n<li>(with Aviezri Fraenkel)\u00a0<a href=\"http:\/\/sites.math.rutgers.edu\/~alexk\/files\/Sierpinski.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">The Sierpinski Sieve of Nim-varieties and Binomial Coefficients<\/a>, Combinatorial number theory, 209&#8211;227, de Gruyter, Berlin, 2007.<\/li>\n<li>(with Mel Nathanson)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/math.NT\/0503177\" target=\"_blank\" rel=\"noopener noreferrer\">Quadratic addition rules for quantum integers<\/a>, Journal of Number Theory, Vol 117, Issue 1, March 2006, pp. 1-13.<\/li>\n<li>(with Steven J. Miller)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/math.NT\/0412003\" target=\"_blank\" rel=\"noopener noreferrer\">Benford&#8217;s Law, values of L-functions and the 3x+1 Problem<\/a>, Acta Arithmetica 120 (2005), 269-297.<\/li>\n<li>(with Yakov Sinai)\u00a0<a href=\"http:\/\/arxiv.org\/abs\/math.NT\/0601622\" target=\"_blank\" rel=\"noopener noreferrer\">Structure Theorem for (d,g,h)-Maps<\/a>, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>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)\u00a0Geometry and &hellip; <a href=\"https:\/\/sites.rutgers.edu\/demo-professor\/research\/\" class=\"\">Read More<\/a><\/p>\n","protected":false},"author":21,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-365","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Research - Henry Rutgers, Ph.D<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/sites.rutgers.edu\/demo-professor\/research\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Research - Henry Rutgers, Ph.D\" \/>\n<meta property=\"og:description\" content=\"Interests Analytic Number Theory; 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