### Abstract

We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.

Original language | English |
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Pages (from-to) | 473-482 |

Number of pages | 10 |

Journal | Russian Journal of Mathematical Physics |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Russian Journal of Mathematical Physics*,

*22*(4), 473-482. https://doi.org/10.1134/S106192081504007X