Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number alpha the Kronecker sequence malpha (m = 1,...,M) is equidistributed modulo one in the limit M --> infinity, and (ii) closed horocycles of length l become equidistributed in the unit tangent bundle T1M of a hyperbolic surface M of finite area, as l --> infinity. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant nu > 0 the Kronecker sequence embedded in T1M along a long closed horocycle becomes equidistributed in T1M for almost all alpha, provided that l = M-nu --> infinity. This equidistribution result holds in fact under explicit diophantine conditions on alpha (e.g. for alpha = root2) 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.