Equidistribution of Kronecker sequences along closed horocycles |
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Authors: | Email author" target="_blank">Jens?MarklofEmail author Andreas?Str?mbergsson |
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Institution: | (1) School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK;(2) Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA;(3) Present address: Department of Mathematics, Uppsala University, Box 480, Uppsala, 75106, Sweden |
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Abstract: | It is well known that (i) for every irrational number the Kronecker
sequence m (m = 1,...,M) is equidistributed modulo one in the
limit
, and (ii) closed horocycles of length
become equidistributed
in the unit tangent bundle
of a hyperbolic surface
of finite area, as
. In the present paper both equidistribution
problems are studied simultaneously: we prove that for any constant
the Kronecker sequence embedded in
along a long closed
horocycle becomes equidistributed in
for almost all , provided
that
. This equidistribution result holds in fact under
explicit diophantine conditions on (e.g. for = 2) provided that
,
with additional assumptions on the Fourier coefficients
of certain automorphic forms. Finally, we show that for
, our
equidistribution theorem implies a recent result of Rudnick and Sarnak
on the uniformity of the pair correlation density of the sequence
n2 modulo one. |
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Keywords: | ((no )) |
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