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22
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 (2 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
Average twin prime conjecture for elliptic curves
, 2007
"... Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s co ..."
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Cited by 20 (6 self)
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Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s conjecture is still widely open. In this paper we prove that Koblitz’s conjecture is true on average over a twoparameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of BarbanDavenportHalberstam,
Sur un problème de Gelfond: la somme des chiffres des nombres premiers
, 2010
"... In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through th ..."
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Cited by 19 (1 self)
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In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the prime numbers is equidistributed modulo 1 if and only if ˛ 2 � n �.
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.
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.
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
TORUS BUNDLES NOT DISTINGUISHED BY TQFT INVARIANTS
"... Abstract. We show that there exist infinitely many pairs of nonhomeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL(2,Z) and its congruence quotients, the classification of SOL (polycyclic) 3mani ..."
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Abstract. We show that there exist infinitely many pairs of nonhomeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL(2,Z) and its congruence quotients, the classification of SOL (polycyclic) 3manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. On the other side we prove that two torus bundles over the circle with the same quantum invariants are (strongly) commensurable. The examples above show that this is the best that it could be expected.
Lagrange’s Theorem and Thin Subsequences of Squares
 CONTRIBUTIONS TO PROBABILITY
"... Probabilistic methods are used to prove that for every E> 0 there exists a sequence A, of squares such that every positive integer is the sum of at most four squares in A, and A,(x) = 0(x=8 + y. ..."
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Cited by 3 (1 self)
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Probabilistic methods are used to prove that for every E> 0 there exists a sequence A, of squares such that every positive integer is the sum of at most four squares in A, and A,(x) = 0(x=8 + y.
Prime pairs and zeta’s zeros
, 2007
"... Abstract. There is extensive numerical support for the primepair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of π2r(x), the number of prime pairs (p, p + 2r) with p ≤ x. However, it is still not known whether there are infinitely many prime pairs with given even diffe ..."
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Abstract. There is extensive numerical support for the primepair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of π2r(x), the number of prime pairs (p, p + 2r) with p ≤ x. However, it is still not known whether there are infinitely many prime pairs with given even difference! Using a strong hypothesis on (weighted) equidistribution of primes in arithmetic progressions, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen. The present author uses a Tauberian approach to derive that the PPC is equivalent to specific boundary behavior of certain functions involving zeta’s complex zeros. Under Riemann’s Hypothesis (RH) and on the real axis these functions resemble paircorrelation expressions. A speculative extension of Montgomery’s classical work (1973) would imply that there must be an abundance of prime pairs. 1.
BOUNDED GAPS BETWEEN PRODUCTS OF PRIMES WITH APPLICATIONS TO IDEAL CLASS GROUPS AND ELLIPTIC CURVES
"... and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their ..."
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and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their prime factors in a set of primes P. Our result holds for any P of positive density that satis es a SiegelWal sz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that P satis es a BombieriVinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of Lfunctions, and triviality of ranks of elliptic curves. 1.