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55
Uniform expansion bound for Cayley graphs of SL2(Fp)
 ANN. MATH
"... We prove that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2,Z) generating a nonelementary subgroup, and with respect to generators chosen at random in SL2(Fp). ..."
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Cited by 89 (11 self)
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We prove that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2,Z) generating a nonelementary subgroup, and with respect to generators chosen at random in SL2(Fp).
Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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Cited by 42 (8 self)
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
A PROBABILISTIC TECHNIQUE FOR FINDING ALMOSTPERIODS IN ADDITIVE COMBINATORICS
"... We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmet ..."
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Cited by 32 (3 self)
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We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmetic progressions in sumsets A +B in Zp. The bounds we obtain for these results are not the best ones known—these being established using Fourier analysis—but they are of a somewhat comparable quality, which is unusual for a method that is completely combinatorial. Furthermore, we are able to find long arithmetic progressions in sets A + B even when both A and B have density close to 1 / logp, which is much sparser than has previously been possible.
Bounding multiplicative energy by the sumset
 Adv. Math
"... Abstract. We prove that the sumset or the productset of any finite set of real numbers, A, is at least A  4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A, A). 1. ..."
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Cited by 32 (1 self)
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Abstract. We prove that the sumset or the productset of any finite set of real numbers, A, is at least A  4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A, A). 1.
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
Growth in SL3(Z/pZ
 J. Eur. Math. Soc
"... Abstract. Let G = SL3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by S  the number of elements of a finite set S. Assume A  < G  1−δ for some δ> 0. Then A · A · A > A  1+ǫ, where ǫ> 0 depends only on δ. Othe ..."
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Cited by 19 (2 self)
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Abstract. Let G = SL3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by S  the number of elements of a finite set S. Assume A  < G  1−δ for some δ> 0. Then A · A · A > A  1+ǫ, where ǫ> 0 depends only on δ. Other results on growth and generation follow. 1.
Product theorems in SL2 and SL3
 Journal of the Institute of Mathematics of Jussieu
"... Abstract We study product theorems for matrix spaces. In particular, we prove the following theorems. Theorem 1. For all ε> 0, there is δ> 0 such that if A ⊂ SL3(Z) is a finite set, then either A intersects a coset of a nilpotent subgroup in a set of size at least A1−ε, or A3 > A1+δ. ..."
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Cited by 17 (0 self)
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Abstract We study product theorems for matrix spaces. In particular, we prove the following theorems. Theorem 1. For all ε> 0, there is δ> 0 such that if A ⊂ SL3(Z) is a finite set, then either A intersects a coset of a nilpotent subgroup in a set of size at least A1−ε, or A3 > A1+δ. Theorem 2. Let A be a finite subset of SL2(C). Then either A is contained in a virtually abelian subgroup, or A3 > cA1+δ for some absolute constant δ> 0. Here A3 = {a1a2a3: ai ∈ A, i = 1, 2, 3} is the 3fold product set of A. §0. Introduction. The aim of this paper is to establish product theorems for matrix spaces, in particular SL2(Z) and SL3(Z). Applications to convolution inequalities will appear in a forthcoming paper. Recall first Tits ’ Alternative for linear groups G over a field of characteristic 0:
Approximate groups, II: The solvable linear case
 Q. J. Math
"... Abstract. We describe the structure of “Kapproximate subgroups ” of solvable subgroups of GLn(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsionfree nilpotent groups [3], we show that such approximate s ..."
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Cited by 14 (7 self)
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Abstract. We describe the structure of “Kapproximate subgroups ” of solvable subgroups of GLn(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsionfree nilpotent groups [3], we show that such approximate subgroups are efficiently controlled by nilpotent progressions.