Density of the multidimensional Lagrange spectrum
Abstract
The Lagrange spectrum is a classical object in number theory, defined as the set of values of liminfq→∞ q dist(qα,ℤ) where α runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.
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BibTeX
@misc{kleinbock2026density,
title = {{Density of the multidimensional Lagrange spectrum}},
author = {Dmitry Kleinbock},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30735},
eprint = {2608.30735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30735},
}
RIS
TY - UNPB
TI - Density of the multidimensional Lagrange spectrum
AU - Dmitry Kleinbock
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.30735
UR - https://arxiv.org/abs/2608.30735
AB - The Lagrange spectrum is a classical object in number theory, defined as the set of values of liminf_q→∞ q dist(qα,ℤ) where α runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.
ER -