 02496 Jens Marklof, Andreas Strombergsson
 Equidistribution of Kronecker sequences along closed horocycles
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Nov 29, 02

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Abstract. It is well known that (i) for every irrational number $\alpha$ the Kronecker sequence $m\alpha$ ($m=1,\ldots,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the unit tangent bundle $\TM$ 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 ${\minexp}>0$ the Kronecker sequence embedded in $\TM$ along a long closed horocycle becomes equidistributed in $\TM$ for almost all $\alpha$, provided
that $\ell=M^{\minexp}\to\infty$. 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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