Equidistribution of Kronecker sequences along closed horocycles

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.