Equidistribution of Kronecker sequences along closed horocycles
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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.
39 pages
39 pages