New Lattice Point Asymptotics for Products of Upper Half-planes

DOI10.1093/IMRN/RNQ120zbMATH Open1315.11040arXiv0904.3020OpenAlexW2040087947MaRDI QIDQ3002513

R. J. Miatello, Roelof Bruggeman, Fritz Grunewald

Publication date: 20 May 2011

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: Let $Gamma$ be an irreducible lattice in $PS{L}_2(RR{)}^d$ ($dinNN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances $$ defined in (1). We prove asymptotic results on the number of $gminGm$ such that $d(z,gammaz$ 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 $gammaz$ such that the hyperbolic distance in each of the factors satisfies $d((gammaz{)}_j,z_j)leT$. The error term, as $Toinfty$ 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 $A_{j}led((gammaz{)}_j,z_j)<B_j$, with fixed $A_j<B_j$ in some factors, while in the remaining factors $0led((gammaz{)}_j,z_j)leT$ is satisfied.

Full work available at URL: https://arxiv.org/abs/0904.3020

zbMATH Keywords

spectral gapspectral measureHilbert modular groupirreducible latticeSelberg transformvector-valued distance

Mathematics Subject Classification ID

Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) (11M36) Spectral theory; trace formulas (e.g., that of Selberg) (11F72)

Related Items (3)

Sums of squares in real quadratic fields and Hilbert modular groups ⋮ Local average in hyperbolic lattice point counting, with an appendix by Niko Laaksonen ⋮ Quantitative ergodic theorems and their number-theoretic applications

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