### 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.

Original language | English |
---|---|

Pages (from-to) | 977-987 |

Number of pages | 11 |

Journal | St. Petersburg Mathematical Journal |

Volume | 23 |

Issue number | 6 |

DOIs | |

Publication status | Published - 28 Dec 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Adiabatic limits
- Convexity
- Domains
- Foliation
- Integer points
- Laplace operator
- Lattices

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics

### Cite this

*St. Petersburg Mathematical Journal*,

*23*(6), 977-987. https://doi.org/10.1090/S1061-0022-2012-01225-2

**Integer points in domains and adiabatic limits.** / Kordyukov, Y. A.; Yakovlev, A. A.

Research output: Contribution to journal › Article

*St. Petersburg Mathematical Journal*, vol. 23, no. 6, pp. 977-987. https://doi.org/10.1090/S1061-0022-2012-01225-2

}

TY - JOUR

T1 - Integer points in domains and adiabatic limits

AU - Kordyukov, Y. A.

AU - Yakovlev, A. A.

PY - 2012/12/28

Y1 - 2012/12/28

N2 - 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.

AB - 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.

KW - Adiabatic limits

KW - Convexity

KW - Domains

KW - Foliation

KW - Integer points

KW - Laplace operator

KW - Lattices

UR - http://www.scopus.com/inward/record.url?scp=84871515619&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871515619&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-2012-01225-2

DO - 10.1090/S1061-0022-2012-01225-2

M3 - Article

AN - SCOPUS:84871515619

VL - 23

SP - 977

EP - 987

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 6

ER -