The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves

By Alex Kontorovich. 2007

49 citations recorded by the source corpus.

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume 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)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta 149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b10fee09ab3460f96a1e3007dbffabaa4b43612b

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Citations in the corpus

Almost prime Pythagorean triples in thin orbits (2010, methodology): “This was achieved with some restrictions in [22], [23], and it is our present goal to improve the results there in a more general setting.”
Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, background): “Our proof of the equidistribution works for hyperbolic surfaces as well, and in particular solves the counting problem in [21] for any norm and without assumptions on the stabilizer.”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, unknown): “Denote by Uλσ and U λ τ the representations of K obtained by restricting U λ |K to the subspace Vσ and Vτ respectively.”