Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Equidistribution of Kronecker sequences along closed horocycles   It is well known that (i) for every irrational number $\alpha$ the Kronecker
sequence $m\alpha$ ($m=1,...,M$) is equidistributed modulo one in the limit
$M\to\infty$, and (ii) closed horocycles of length $\ell$ become
equidistributed in the unit tangent bundle $T_1 M$ of a hyperbolic surface $M$
of finite area, as $\ell\to\infty$. In the present paper both equidistribution
problems are studied simultaneously: we prove that for any constant $\nu > 0$
the Kronecker sequence embedded in $T_1 M$ along a long closed horocycle
becomes equidistributed in $T_1 M$ for almost all $\alpha$, provided that $\ell
= M^{\nu} \to \infty$. This equidistribution result holds in fact under
explicit diophantine conditions on $\alpha$ (e.g., for $\alpha=\sqrt 2$)
provided that $\nu<1$, or $\nu<2$ with additional assumptions on the Fourier
coefficients of certain automorphic forms. Finally, we show that for $\nu=2$,
our equidistribution theorem implies a recent result of Rudnick and Sarnak on
the uniformity of the pair correlation density of the sequence $n^2 \alpha$
modulo one.

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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex