Integer points in domains and adiabatic limits
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Y. A. Kordyukov and A. A. Yakovlev
Translated by: the authors - St. Petersburg Math. J. 23 (2012), 977-987
- DOI: https://doi.org/10.1090/S1061-0022-2012-01225-2
- 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.References
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Bibliographic Information
- Y. A. Kordyukov
- Affiliation: Institute of Mathematics, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia
- MR Author ID: 227886
- ORCID: 0000-0003-2957-2873
- 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
- Received by editor(s): June 25, 2010
- Published electronically: September 17, 2012
- Additional Notes: Supported by RFBR (grant no. 09-01-00389)
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 977-987
- MSC (2010): Primary 11P21; Secondary 58J50
- DOI: https://doi.org/10.1090/S1061-0022-2012-01225-2
- MathSciNet review: 2962181