Lattice points for products of upper half planes
| dc.creator | Bruggeman, Roelof | |
| dc.creator | Grunewald, Fritz | |
| dc.creator | Miatello, Roberto | |
| dc.date | 2009-04-20 | |
| dc.date.accessioned | 2026-07-07T13:06:20Z | |
| dc.date.available | 2026-07-07T13:06:20Z | |
| dc.description | Let $Γ$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1). We prove asymptotic results on the number of $\gm\in\Gm$ such that $d(z,γz$ is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes % improve the error terms improve the existing error terms (by Gorodnick and Nevo) and generalize the Selberg error term for $d=1$. We give an asymptotic formula for the number of lattice points $γz$ such that the hyperbolic distance in each of the factors satisfies $d((γz)_j, z_j)\le T$. The error term, as $T \to \infty$ generalizes the error term given by Selberg for $d=1$, also we describe how the counting function depends on $z$. We also prove asymptotic results when the distance satisfies $A_j \le d((γz)_j, z_j) < B_j$, with fixed $A_j < B_j$ in some factors, while in the remaining factors $0 \le d((γz)_j, z_j) \le T$ is satisfied. | |
| dc.identifier | https://arxiv.org/abs/0904.3020 | |
| dc.identifier | http://arxiv.org/abs/0904.3020 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/227747 | |
| dc.subject | Number Theory | |
| dc.subject | 11F41 (Primary), 11F72 (Secondary) | |
| dc.title | Lattice points for products of upper half planes | |
| dc.type | text |