Equidistribution of Kronecker sequences along closed horocycles

2002/11/12 by Jens Marklof, Marklof, Jens, Andreas Strombergsson +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP) #math.NT #math.SP

paper · pdf · doi:10.48550/arxiv.math/0211189

39 pages

arxiv created 2002/11/12 · arxiv updated 2009/11/30

Abstract

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 T₁ M 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 T₁ M along a long closed horocycle becomes equidistributed in T₁ M for almost all α, provided that ℓ = M^ν → ∞. 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 n² α modulo one.

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