Equidistribution of Kronecker sequences along closed horocycles

By Jens Marklof, Andreas Strombergsson. 2002

49 citations recorded by the source corpus.

AbstractIt is well known that (i) for every irrational number α the Kronecker sequence mα (m = 1,...,M) is equidistributed modulo one in the limit $$ \rightarrow\infty $$ , and (ii) closed horocycles of length $$ \ell $$ become equidistributed in the unit tangent bundle $$ T_1\mathcal{M} $$ of a hyperbolic surface $$ \mathcal{M} $$ of finite area, as $$ \ell\rightarrow\infty $$ . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $$ \nu > 0 $$ the Kronecker sequence embedded in $$ T_1\mathcal{M} $$ along a long closed horocycle becomes equidistributed in $$ T_1\mathcal{M} $$ for almost all α, provided that $$ $$ . This equidistribution result holds in fact under explicit diophantine conditions on α (e.g. for α = √2) provided that $$ \nu < 1 $$ , $$ \nu < 2 $$ 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 n2α modulo one.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/051373d8b96f5302860a6c746218588a16229b07

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In Algebra and Number Theory

  1. Igor E. Shparlinski
  2. Jens Marklof
  3. Andreas Strombergsson

In Mathematical Physics

  1. Igor E. Shparlinski
  2. Daniel El-Baz
  3. Ilya Vinogradov
  4. Jens Marklof
  5. Andreas Strombergsson

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