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Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Cited by 30 (3 self)
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
GENERALIZATION OF SELBERG’S 3/16 THEOREM AND AFFINE SIEVE
"... A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper ..."
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Cited by 25 (3 self)
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A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper
The hyperbolic lattice point count in infinite volume with applications to sieves
 arXive:0712.139, 2008. CIRCLE PACKING 50
"... Abstract. We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinitevolume hyperbolic manifolds, with error terms which are uniform as the lattice moves through “congruence ” subgroups. We give the following application to ..."
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Cited by 18 (5 self)
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Abstract. We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinitevolume hyperbolic manifolds, with error terms which are uniform as the lattice moves through “congruence ” subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c, d) = c2 + d2, but restrict (c, d) to the orbit O = (0, 1)Γ, where Γ is an infiniteindex nonelementary finitelygenerated subgroup of SL(2, Z) containing unipotent elements. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R> 4/(δ−θ), where θ> 1/2 is the spectral gap and δ < 1 is the Hausdorff dimension of the limit set of Γ. If δ> 149/150, then we can take θ = 5/6, giving R = 25. The limit of this method is R = 9 for δ − θ> 4/9. This is the same number of prime factors as attained in Brun’s original attack on the twin prime conjecture. 1.
On representations of integers in thin subgroups of SL2(Z
 Geom. Funct. Anal
"... Abstract. Let Γ < SL(2,Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v0, w0 ∈ Z2 \ {0}. We consider the set S of all integers occurring in v0γ tw0, for γ ∈ Γ. Assume that the limit set of Γ has Hausdorff dimension δ> 0.9999 ..."
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Cited by 17 (7 self)
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Abstract. Let Γ < SL(2,Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v0, w0 ∈ Z2 \ {0}. We consider the set S of all integers occurring in v0γ tw0, for γ ∈ Γ. Assume that the limit set of Γ has Hausdorff dimension δ> 0.99995, that is, Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6th spectral gap in infinitevolume, we show that S contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set E(N) of integers n  < N which are locally admissible (n ∈ S(mod q) for all q ≥ 1) but fail to be globally represented, n / ∈ S, has a power savings, E(N)  N1−ε0
Integral Apollonian Packings
"... Abstract. We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed ..."
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Cited by 13 (1 self)
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Abstract. We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed so that they are mutually tangent. This configuration is unique up to rigid motions. As far as I can tell there is no official exact size for these coins, but the diameters of 24, 21, Figure 1. and 18 millimeters are accurate to the nearest millimeter and I assume henceforth that these are the actual diameters. Let C be the unique (see below) circle that is tangent to the three coins as shown in Figure 2. It is a small coincidence that its diameter is rational, as indicated. d = diameter d 2 = 21 mm d 3 = 24 mm C d 4 = 504 mm
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.
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
 PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2010
"... We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpin ..."
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Cited by 11 (4 self)
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We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.
Approximate groups and their applications: work of Bourgain
 Gamburd, Helfgott, and Sarnak. Current Events Bulletin, AMS
"... Abstract. This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion ..."
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Cited by 10 (2 self)
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Abstract. This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of FrĕımanRuzsa, BourgainKatzTao, Helfgott and others in which the structure of such objects is elucidated. We then move on to the applications. In particular we will look at the work of Bourgain and Gamburd on expansion properties of Cayley graphs on SL2(Fp) and at its application in the work of Bourgain, Gamburd and Sarnak on nonlinear sieving problems. 1.