Equidistribution of Kronecker sequences along closed horocycles
| dc.creator | Marklof, Jens | |
| dc.creator | Strombergsson, Andreas | |
| dc.date | 2002-11-12 | |
| dc.date.accessioned | 2026-07-07T04:52:51Z | |
| dc.date.available | 2026-07-07T04:52:51Z | |
| dc.description | It is well known that (i) for every irrational number $α$ the Kronecker sequence $mα$ ($m=1,...,M$) 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 $ν> 0$ the Kronecker sequence embedded in $T_1 M$ along a long closed horocycle becomes equidistributed in $T_1 M$ for almost all $α$, provided that $\ell = M^ν \to \infty$. This equidistribution result holds in fact under explicit diophantine conditions on $α$ (e.g., for $α=\sqrt 2$) provided that $ν<1$, or $ν<2$ with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for $ν=2$, our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence $n^2 α$ modulo one. | |
| dc.description | 39 pages | |
| dc.identifier | https://arxiv.org/abs/math/0211189 | |
| dc.identifier | http://arxiv.org/abs/math/0211189 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65625 | |
| dc.subject | Number Theory | |
| dc.subject | Spectral Theory | |
| dc.title | Equidistribution of Kronecker sequences along closed horocycles | |
| dc.type | text |