Equidistribution of Kronecker Sequences along Closed Horocycles

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 → ∞, 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 → ∞. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant ν > 0 the Kronecker sequence embedded in T1M along a long closed horocycle becomes equidistributed in T1M for almost all α, provided that l = M → ∞. This equidistribution result holds in fact under explicit diophantine conditions on α (e.g., for α = √ 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 nα modulo one.