Abstract
We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assumptions on the boundary of the domain. We study the average remainder estimates, where the averages are taken over rotated images of the domain by a subgroup of the group $$\hbox {SO}(n)$$SO(n) of orthogonal transformations of the Euclidean space $${\mathbb {R}}^n$$Rn. Using these results, we improve the remainder estimate in the adiabatic limit formula 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 when 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) | 97-111 |
| Number of pages | 15 |
| Journal | Monatshefte fur Mathematik |
| Volume | 178 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 1 Sep 2015 |
| Externally published | Yes |
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Keywords
- Adiabatic limits
- Anisotropically expanding domains
- Convexity
- Foliation
- Integer points
- Laplace operator
ASJC Scopus subject areas
- Mathematics(all)
Cite this
The number of integer points in a family of anisotropically expanding domains. / Kordyukov, Yuri A.; Yakovlev, Andrey A.
In: Monatshefte fur Mathematik, Vol. 178, No. 1, 01.09.2015, p. 97-111.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The number of integer points in a family of anisotropically expanding domains
AU - Kordyukov, Yuri A.
AU - Yakovlev, Andrey A.
PY - 2015/9/1
Y1 - 2015/9/1
N2 - We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assumptions on the boundary of the domain. We study the average remainder estimates, where the averages are taken over rotated images of the domain by a subgroup of the group $$\hbox {SO}(n)$$SO(n) of orthogonal transformations of the Euclidean space $${\mathbb {R}}^n$$Rn. Using these results, we improve the remainder estimate in the adiabatic limit formula 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 when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.
AB - We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assumptions on the boundary of the domain. We study the average remainder estimates, where the averages are taken over rotated images of the domain by a subgroup of the group $$\hbox {SO}(n)$$SO(n) of orthogonal transformations of the Euclidean space $${\mathbb {R}}^n$$Rn. Using these results, we improve the remainder estimate in the adiabatic limit formula 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 when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.
KW - Adiabatic limits
KW - Anisotropically expanding domains
KW - Convexity
KW - Foliation
KW - Integer points
KW - Laplace operator
UR - http://www.scopus.com/inward/record.url?scp=84936816751&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84936816751&partnerID=8YFLogxK
U2 - 10.1007/s00605-015-0787-7
DO - 10.1007/s00605-015-0787-7
M3 - Article
AN - SCOPUS:84936816751
VL - 178
SP - 97
EP - 111
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
SN - 0026-9255
IS - 1
ER -