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Hee Oh

Publications and source records attributed to Hee Oh.

At least 73 records · Page 4Linked to original sources
arXiv2015

Expanding maps and continued fractions

We obtain a power saving in the error term for a semigroup congruence lattice point count related to continued fractions. This is done by adapting arguments from recent work of Oh and Winter (2014) that give uniform bounds for certain transfer operators in the congruence aspect. Our arguments also build crucially on work of Naud (2005) and Bourgain, Gamburd and Sarnak (2011). The result we obtain, together with a certain conjecture about the multiplicative combinatorics of $\mathrm{SL}_2(\mathbf{Z})$ that we highlight in the sequel, can be used to obtain an improvement on the size of the exceptional set in Bourgain and Kontorovich's work (2014) on Zaremba's conjecture.

math.NT↗
arXiv2014

Harmonic analysis, Ergodic theory and Counting for thin groups

For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.

math.NT↗
arXiv2014

Apollonian circle packings: Dynamics and Number theory

We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the fall of 2013.

math.DS↗
arXiv2013

Matrix coefficients, Counting and Primes for orbits of geometrically finite groups

Let G:=SO(n,1)^\circ and Γbe a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(Γ\ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and for delta>n-2 for n>= 4, we obtain an {\it effective} archimedean counting result for a discrete orbit of Γin a homogeneous space H\G, where H is the trivial group, an affine symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {B_T} of compact subsets in H\G, there exists η>0 such that #[e]\G\cap B_T=M(B_T) +O(M(B_T)^{1-η}) for an explicit measure M on H\G, which depends on Gamma. We also apply affine sieve and describe the distribution of almost primes on orbits of Γin arithmetic settings. One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of L^2(Γ\ G) that we prove by combining methods from spectral analysis, harmonic analysis and ergodic theory. An effective mixing for the Bowen-Margulis-Sullivan measure is also obtained as an application of our methods.

math.NT↗
arXiv2013

Eigenvalues of congruence covers of geometrically finite hyperbolic manifolds

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(λ_0)>0.

math.SP↗
arXiv2012

Effective Circle Count for Apollonian packings and Closed horospheres

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.

math.DS↗
arXiv2012

Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approach is the description for the limiting distribution of the orthogonal translates of a totally geodesically immersed closed submanifold of Gamma\H^n. We also give a criterion on the finiteness of the Gamma skinning size of w_0 for Gamma geometrically finite.

math.DS↗
arXiv2012

Effective Equidistribution of Closed horocycles for Geometrically finite surfaces

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of Sarnak in 1981 for surfaces of finite area. We also discuss its applications in Affine sieves.

math.DS↗
arXiv2011

Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids

For any non-uniform lattice $Γ$ in $SL(2,R)$, we describe the limit distribution of orthogonal translates of a divergent geodesic in $Γ\backslash SL(2,R)$. As an application, for a quadratic form $Q$ of signature $(2,1)$, a lattice $Γ$ in its isometry group, and $v_0\in R^3$ with $Q(v_0)>0$, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit $v_0Γ$ of norm at most $T$, when the stabilizer of $v_0$ in $Γ$ is finite. Our result in particular implies that for any non-zero integer $d$, the smoothed count for number of integral binary quadratic forms with discriminant $d^2$ and with coefficients bounded by $T$ is asymptotic to $c\cdot T \log T +O(T)$.

math.NT↗
arXiv2011

The Asymptotic distribution of circles in the orbits of Kleinian groups

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.

math.DS↗
arXiv2010

Rational points on homogeneous varieties and Equidistribution of Adelic periods

Let U:=L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively on U(K_v) for almost all places v of K, we obtain the asymptotic of the number of rational points in U(K) with height bounded by T, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.

math.AG↗
arXiv2010

Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one.

math.DS↗
arXiv2010

On the distribution of orbits of geometrically finite hyperbolic groups on the boundary (with appendix by Francois Maucourant)

We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $Γ$ admits a finite Bowen-Margulis-Sullivan measure (for instance, if it is geometrically finite), then every $Γ$-orbit in the hyperbolic space is equidistributed with respect to the Patterson-Sullivan measure supported on the limit set of $Γ$.The appendix by Maucourant is the extension of a part of his thesis where he obtains the same result as a simple application of Roblin's theorem. Our approach is via establishing the equidistribution of solvable flows on the unit tangent bundle of the quotient manifold, which is of independent interest.

math.DS↗
arXiv2010

Almost prime Pythagorean triples in thin orbits

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.

math.NT↗
arXiv2008

Integral points on symmetric varieties and Satake compatifications

Let V be an affine symmetric variety defined over Q. We compute the asymptotic distribution of the angular components of the integral points in V. This distribution is described by a family of invariant measures concentrated on the Satake boundary of V. In the course of the proof, we describe the structure of the Satake compactifications for general affine symmetic varieties and compute the asymptotic of the volumes of norm balls.

math.NT↗
arXiv2008

Manin's and Peyre's conjectures on rational points and adelic mixing

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K. We prove Manin's conjecture on the asymptotic (as T\to \infty) of the number of K-rational points of X of height less than T, and give an explicit construction of a measure on X(A), generalizing Peyre's measure, which describes the asymptotic distribution of the rational points G(K) on X(A). Our approach is based on the mixing property of L^2(G(K)\G(A)) which we obtain with a rate of convergence.

math.NT↗