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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Integer points in domains and adiabatic limits


Authors: Y. A. Kordyukov and A. A. Yakovlev
Translated by: the authors
Original publication: Algebra i Analiz, tom 23 (2011), nomer 6.
Journal: St. Petersburg Math. J. 23 (2012), 977-987
MSC (2010): Primary 11P21; Secondary 58J50
Published electronically: September 17, 2012
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Abstract: An asymptotic formula is proved for the number of integral points in a family of bounded domains with smooth boundary in Euclidean space; these domains remain unchanged along some linear subspace and expand in the directions orthogonal to this subspace. A sharper estimate for the remainder is obtained in the case where the domains are strictly convex. These results make it possible to improve the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in the particular case where the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.


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Additional Information

Y. A. Kordyukov
Affiliation: Institute of Mathematics, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia
Email: yurikor@matem.anrb.ru

A. A. Yakovlev
Affiliation: Department of Mathematics, Ufa State Aviation Technical University, K. Marx str. 12, Ufa 450000, Russia
Email: yakovlevandrey@yandex.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01225-2
PII: S 1061-0022(2012)01225-2
Keywords: Integer points, lattices, domains, convexity, adiabatic limits, foliation, Laplace operator
Received by editor(s): June 25, 2010
Published electronically: September 17, 2012
Additional Notes: Supported by RFBR (grant no. 09-01-00389)
Article copyright: © Copyright 2012 American Mathematical Society