Hee Oh

Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants

Let k be a local field and G the group of k-rational points of a connected reductive linear algebraic group over k with k-semisimple rank(G) ≥ 2. Let K be a good maximal compact subgroup of G. For a unitary representation ρ of G, a vector v in ρ is called Kfinite if the subspace spanned byKv is finite dimensional. We will use the termK-matrix coefficients (resp. K-finite matrix coefficients) of ρ to refer to its matrix coefficients with respect to K-invariant (resp. K-finite) unit vectors. Following [BT], we denote by G the subgroup generated by the unipotent k-split subgroups of G. The main goal of the present paper is to construct a class of uniform pointwise bounds for the K-finite matrix coefficients of all infinite dimensional irreducible unitary representations of G, or more generally of all unitary representations of G without a non-zero G-invariant vector. Let A be a maximal k-split torus and A the closed positive Weyl chamber of A such that the Cartan decomposition G = KAΩK holds where Ω is a finite subset of the

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.

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.

Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Let X be a symmetric space of noncompact type, and let Γ be a lattice in the isometry group of X. We study the distribution of orbits of Γ acting on the symmetric space X and its geometric boundary X(∞), generalizing the main equidistribution result of Margulis’s thesis [M, Theorem 6] to higher-rank symmetric spaces. More precisely, for any y ∈ X and b ∈ X(∞), we investigate the distribution of the set {(yγ, bγ^(−1)) : γ ∈ } in X × X(∞). It is proved, in particular, that the orbits of Γ in the Furstenberg boundary are equidistributed and that the orbits of Γ in X are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah’s result [S, Corollary 1.2] based on Ratner’s measure-classification theorem [R1, Theorem 1].

Ergodic theoretic proof of equidistribution of Hecke points

We prove the equidistribution of Hecke points for any connected non-compact

Almost prime Pythagorean triples in thin orbits

\mathbb{Q}

Equidistribution of Integer Points on a Family of Homogeneous Varieties: A Problem of Linnik

-simple real algebraic group G and an arithmetic subgroup

Matrix coefficients, counting and primes for orbits of geometrically finite groups

\Gamma\subset G(\mathbb{Q})

Integral points on symmetric varieties and Satake compatifications

, generalizing a theorem of Clozel, Oh and Ullmo. The main tool is a theorem of Mozes and Shah on unipotent flows.

Strong wavefront lemma and counting lattice points in sectors

Abstract For the ternary quadratic form Q(x) = x2 + y2 − z2 and a non-zero Pythagorean triple x0 ∈ ℤ3 lying on the cone Q(x) = 0, we consider an orbit 𝒪 = x0Γ of a finitely generated subgroup Γ < SOQ(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.