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Publications Influence

Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

- Alex Kontorovich, H. Oh
- Mathematics
- 14 November 2008

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… Expand

105 9- PDF

On Zaremba's conjecture

- J. Bourgain, Alex Kontorovich
- Mathematics
- 1 July 2014

Zaremba’s 1971 conjecture predicts that every integer appears as the denominator of a nite continued fraction whose partial quotients are bounded by an absolute constant. We conrm this conjecture for… Expand

46 9

Sector Estimates for Hyperbolic Isometries

- J. Bourgain, Alex Kontorovich, P. Sarnak
- Mathematics
- 25 January 2010

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… Expand

45 8- PDF

On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions

- D. Goldfeld, Alex Kontorovich
- Mathematics
- 29 March 2012

We present an explicit approach to the GL(3) Kuznetsov formula. As an application, for a restricted class of test functions, we obtain the low-lying zero densities for the following three families:… Expand

26 5- PDF

On the Pair Correlation Density for Hyperbolic Angles

- Dubi Kelmer, Alex Kontorovich
- Mathematics
- 3 August 2013

Let $\Gamma< \mathrm{PSL}_2(\mathbb{R})$ be a lattice and $\omega\in \mathbb{H}$ a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density… Expand

12 4- PDF

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

- J. Bourgain, Alex Kontorovich
- Mathematics
- 25 January 2010

Let Gamma , for gamma in Gamma. Assume that the limit set of Gamma has Hausdorff dimension delta>0.99995, that is, Gamma is thin but not too thin. Using a variant of the circle method, new bilinear… Expand

23 3- PDF

On the Determination of the Plancherel measure for Lebedev-Whittaker transforms on GL(n)

- D. Goldfeld, Alex Kontorovich
- Mathematics
- 24 February 2011

We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n).

17 3- PDF

On Zarembaʼs conjecture

- J. Bourgain, Alex Kontorovich
- Mathematics
- 2 March 2011

Abstract It is shown that there is a constant A and a density one subset S of the positive integers such that, for each q ∈ S , there is some 1 ⩽ p q , ( p , q ) = 1 , so that p q has all its partial… Expand

27 2- PDF

The Sierpiński Sieve of Nim-varieties and Binomial Coefficients

- A. Fraenkel, Alex Kontorovich
- Mathematics
- 2006

We consider the variety XY , where instead of multiplication, we take the Nim-product. Its geometry turns out to be the Sierpinski sieve, which is well known to be connected to Pascal’s triangle… Expand

14 2- PDF

Stochastic Models for the 3x+1 and 5x+1 Problems

- Alex Kontorovich, J. Lagarias
- Mathematics
- 11 October 2009

This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem. The stochastic models are… Expand

11 2- PDF