PreprintCC BY-NC-ND 4.0arXiv
Distribution of points near the origin in the d-dimensional Lagrange spectrum
Abstract
We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension d≥ 2. We show the Hausdorff dimension of the set of points in ℝd whose best approximation constant lies in [ε,ε(1+δεd)], for some constant δ>0, is positive, and for a slightly larger set approaches full dimension as ε → 0. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. We also use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.
Related
- Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum
- Density of the multidimensional Lagrange spectrum
- Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half
- Existence of a Hall ray in higher dimensional Lagrange spectra
- Lenses make many points: Pure point spectrum disproves Simon's L2 conjecture
BibTeX
@misc{ward2026distribution,
title = {{Distribution of points near the origin in the $d$-dimensional Lagrange spectrum}},
author = {Benjamin Ward},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05311},
eprint = {2609.05311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05311},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Distribution of points near the origin in the d-dimensional Lagrange spectrum
AU - Benjamin Ward
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.05311
UR - https://arxiv.org/abs/2609.05311
AB - We develop a new framework, inspired by Schmidt's games, to study the Lagrange spectrum for simultaneous Diophantine approximation in dimension d≥ 2. We show the Hausdorff dimension of the set of points in ℝ^d whose best approximation constant lies in [ε,ε(1+δε^d)], for some constant δ>0, is positive, and for a slightly larger set approaches full dimension as ε → 0. The proof combines a novel application of the Simplex lemma near rational points with the game-theoretic framework. We also use an elementary observation to show that the naturally defined Lagrange spectrum for systems of linear forms is uncountable in the case of square matrices.
ER -