TY - JOUR

T1 - The hyperbolic lattice point count in infinite volume with applications to sieves

AU - Kontorovich, Alex V.

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2009/7

Y1 - 2009/7

N2 - We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms that 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{script} = (0, 1)Γ,whereΓ is an infinite-index, nonelementary, finitely generated subgroup of SL(2, Z{double-struck}). Assume that the Reimann surface Γ\H{double-struck} has a cusp at infinity. We show that the set of values f (O{script}) contains infinitely many integers having at most R prime factors for any R>4/(δ - Θ),whereΘ>1/2 is the spectral gap and δ<1 is the Hausdorff dimension of the limit set of Γ.Ifδ>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.

AB - We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms that 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{script} = (0, 1)Γ,whereΓ is an infinite-index, nonelementary, finitely generated subgroup of SL(2, Z{double-struck}). Assume that the Reimann surface Γ\H{double-struck} has a cusp at infinity. We show that the set of values f (O{script}) contains infinitely many integers having at most R prime factors for any R>4/(δ - Θ),whereΘ>1/2 is the spectral gap and δ<1 is the Hausdorff dimension of the limit set of Γ.Ifδ>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.

UR - http://www.scopus.com/inward/record.url?scp=77953584622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953584622&partnerID=8YFLogxK

U2 - 10.1215/00127094-2009-035

DO - 10.1215/00127094-2009-035

M3 - Article

AN - SCOPUS:77953584622

VL - 149

SP - 1

EP - 36

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -