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

Abstract

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, $f(c,d)=c^2+d^2$, but restrict $(c,d)$ to the orbit $O=(0,1)\Gamma$, where $\Gamma$ is an infinite-index, nonelementary, finitely generated subgroup of $\mathrm{SL}(2,\mathbb{Z})$. Assume that the Reimann surface $\Gamma \backslash \mathbb{H}$ 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 $\delta <1$ is the Hausdorff dimension of the limit set of $\Gamma$. If $\delta >149/150$, 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