Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lattice points for products of upper half planes   Let $\Gamma$ 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 $d(z,\gamma z$ 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 $\gamma z$ such that the hyperbolic distance in each of the
factors satisfies $d((\gamma z)_j, z_j)\le T$. 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 $A_j \le d((\gamma z)_j, z_j) < B_j$, with fixed $A_j <
B_j$ in some factors, while in the remaining factors $0 \le d((\gamma z)_j,
z_j) \le T$ is satisfied.

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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol