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On the localglobal conjecture for integral Apollonian gaskets
 INVENTIONES MATHEMATICAE
, 2013
"... We prove that a set of density one satisfies the localglobal conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket. ..."
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We prove that a set of density one satisfies the localglobal conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.
Effective circle count for Apollonian packings and closed horospheres
, 2012
"... 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 3manifold, whos ..."
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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 3manifold, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.
HARMONIC ANALYSIS, ERGODIC THEORY AND COUNTING FOR THIN GROUPS
, 2012
"... For a geometrically finite group Γ of G = SO(n, 1), we survey recent developments on counting and equidistribution problems for orbits of Γ 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 ..."
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For a geometrically finite group Γ of G = SO(n, 1), we survey recent developments on counting and equidistribution problems for orbits of Γ 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.
FROM APOLLONIUS TO ZAREMBA: LOCALGLOBAL PHENOMENA IN THIN ORBITS
"... Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the localglobal problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractio ..."
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Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the localglobal problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.
Notes on thin matrix groups
"... These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group ..."
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These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.
Apollonian Circles with Integer Curvatures
"... Given four mutually tangent circles (one of them internally tangent to the other three), we can inscribe into each of the remaining curvilinear triangles a unique circle. Continuing iteratively in this manner, we obtain what is known as an Apollonian circle packing. If the initial four circles posse ..."
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Given four mutually tangent circles (one of them internally tangent to the other three), we can inscribe into each of the remaining curvilinear triangles a unique circle. Continuing iteratively in this manner, we obtain what is known as an Apollonian circle packing. If the initial four circles possess integer curvatures (reciprocal radii), then all of the circles in the packing possess integer curvatures. Some introductory accounts of this subject include [1, 2, 3, 4]. We examine just two examples, the first starting with curvatures {−1 2 2 3} (Figure 1) and the second starting with curvatures {−11 21 24 28} (Figure 2). The outer circle is given negative curvature — indicating that the other circles are in its interior — and it is the unique circle with this property. How are the integer curvatures obtained for each example? Define four 4 × 4 matrices 1 =
ORBITAL COUNTING OF CURVES ON ALGEBRAIC SURFACES AND SPHERE PACKINGS
"... Abstract. We realize the Apollonian group associated to an integral Apollonian circle packings, and some of its generalizations, as a group of automorphisms of an algebraic surface. Borrowing some results in the theory of orbit counting, we study the asymptotic of the growth of degrees of elements ..."
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Abstract. We realize the Apollonian group associated to an integral Apollonian circle packings, and some of its generalizations, as a group of automorphisms of an algebraic surface. Borrowing some results in the theory of orbit counting, we study the asymptotic of the growth of degrees of elements in the orbit of a curve on an algebraic surface with respect to a geometrically finite group of its automorphisms. 1.
Apollonian Equilateral Triangles
, 2012
"... Given an equilateral triangle with a the square of its side length and a point in its plane with b, c, d the squares of the distances from the point to the vertices of the triangle, it can be computed that a, b, c, d satisfy 3(a2+b2+ c2+d2) = (a+b+ c+d)2. This paper derives properties of quadruples ..."
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Given an equilateral triangle with a the square of its side length and a point in its plane with b, c, d the squares of the distances from the point to the vertices of the triangle, it can be computed that a, b, c, d satisfy 3(a2+b2+ c2+d2) = (a+b+ c+d)2. This paper derives properties of quadruples of nonnegative integers (a, b, c, d), called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating (a, b, c, a+ b+ c − d) from (a, b, c, d) preserves this feature and that it and analogous ones for the other elements can be represented by four matrices. We examine in detail the triangle group, the group with these operations as generators, and completely classify the orbits of quadruples with respect to the triangle group action. We also compute the number of triangle quadruples generated after a certain number of operations and approximate the number of quadruples bounded by characteristics such as the maximal element. Finally, we prove that the triangle group is a hyperbolic Coxeter group and derive information about the elements of triangle quadruples by invoking Lie groups. We also generalize the problem to higher dimensions. 1
FROM APOLLONIAN PACKINGS TO HOMOGENEOUS SETS
"... Abstract. We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd’s theorem (relating th ..."
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Abstract. We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd’s theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpiński carpets that are Julia sets of hyperbolic rational maps. 1.