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 |
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Keywords
- Adiabatic limits
- Convexity
- Domains
- Foliation
- Integer points
- Laplace operator
- Lattices
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Applied Mathematics
Cite this
Integer points in domains and adiabatic limits. / Kordyukov, Y. A.; Yakovlev, A. A.
In: St. Petersburg Mathematical Journal, Vol. 23, No. 6, 28.12.2012, p. 977-987.Research output: Contribution to journal › Article
}
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 -