Alex Kontorovich

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.

Benford's law, values of L-functions and the 3x+1 problem

We show the leading digits of a variety of systems satisfying certain conditions follow Benfords Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford.

Sector Estimates for Hyperbolic Isometries

We prove various orbital counting statements for Fuchsian groups of the second kind. These are of independent interest, and also are used in the companion paper [BK] to produce primes in the Affine Linear Sieve.

Almost prime Pythagorean triples in thin orbits

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.

On the pair correlation density for hyperbolic angles

Let

Effective equidistribution of shears and applications

\Gamma< \mathrm{PSL}_2(\mathbb{R})

A Pseudo Twin Primes Theorem

be a lattice and

On Zaremba's conjecture

\omega\in \mathbb{H}

On representations of integers in thin subgroups of SL(2,Z)

a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice

On the local-global conjecture for integral Apollonian gaskets

\Gamma \omega