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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 ofExpand
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On Zaremba's conjecture
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 forExpand
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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 AffineExpand
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On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions
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
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On the Pair Correlation Density for Hyperbolic Angles
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 densityExpand
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On representations of integers in thin subgroups of SL(2,Z)
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 bilinearExpand
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On the Determination of the Plancherel measure for Lebedev-Whittaker transforms on GL(n)
We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n).
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On Zarembaʼs conjecture
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 partialExpand
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The Sierpiński Sieve of Nim-varieties and Binomial Coefficients
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 triangleExpand
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Stochastic Models for the 3x+1 and 5x+1 Problems
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 areExpand
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