%0 Generic %1 Marklof2002 %A Marklof, Jens %A Strombergsson, Andreas %D 2002 %K horocycle modular %T Equidistribution of Kronecker sequences along closed horocycles %U http://arxiv.org/abs/math/0211189 %X 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 $M\toınfty$, 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 $\ell\toınfty$. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $> 0$ 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 $\ell = M^\nu ınfty$. This equidistribution result holds in fact under explicit diophantine conditions on $\alpha$ (e.g., for $\alpha=2$) 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.

@misc{Marklof2002, abstract = { 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. }, added-at = {2011-04-26T14:54:37.000+0200}, author = {Marklof, Jens and Strombergsson, Andreas}, biburl = {https://www.bibsonomy.org/bibtex/21caef7a059e33047398377d4243b3812/uludag}, description = {Equidistribution of Kronecker sequences along closed horocycles}, interhash = {c0b926b4f272a37dc150490158cd5c87}, intrahash = {1caef7a059e33047398377d4243b3812}, keywords = {horocycle modular}, note = {cite arxiv:math/0211189 Comment: 39 pages}, timestamp = {2011-04-26T14:54:37.000+0200}, title = {Equidistribution of Kronecker sequences along closed horocycles}, url = {http://arxiv.org/abs/math/0211189}, year = 2002 }