Porosity in Diophantine approximation
Abstract
We provide a general framework for obtaining upper bounds on the packing dimension of certain sets arising from Diophantine approximation using the notion of porosity from geometric measure theory. In particular, the framework relies on showing that such sets are mean porous, as introduced by Koskela and Rohde (Math. Ann., 1997). As applications of our framework we show that: For ε>0 sufficiently small, the set of d-dimensional ε-badly approximable points has packing codimension comparable to εd up to multiplicative constant. Any C2 submanifold of ℝd with bounded first and second partial derivatives intersected with the set of d-dimensional ε-badly approximable points has packing dimension strictly less than the dimension of the submanifold. For any matrix A∈ ℝm× n satisfying certain Diophantine conditions, the set of points that avoid the ε |q|∞−n/m-neighbourhood of the orbit {Aq mod ℤm:q ∈ ℤn} has packing codimension greater than CA εm for CA a constant depending on A and m only. \end{itemize}
Related
- Metric Diophantine approximation on planar and Veronese curves
- Extreme value laws for intrinsic Diophantine approximation on spheres
- Diophantine approximation by primes and Landau--Siegel zeros
- On a Notion of Density Related to Restricted Diophantine Approximation
- Twisted Diophantine Approximation I: Asymptotic Theory
BibTeX
@misc{koivusalo2026porosity,
title = {{Porosity in Diophantine approximation}},
author = {Henna Koivusalo and Benjamin Ward},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00257},
eprint = {2611.00257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00257},
}
RIS
TY - UNPB
TI - Porosity in Diophantine approximation
AU - Henna Koivusalo
AU - Benjamin Ward
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00257
UR - https://arxiv.org/abs/2611.00257
AB - We provide a general framework for obtaining upper bounds on the packing dimension of certain sets arising from Diophantine approximation using the notion of porosity from geometric measure theory. In particular, the framework relies on showing that such sets are mean porous, as introduced by Koskela and Rohde (Math. Ann., 1997). As applications of our framework we show that: For ε>0 sufficiently small, the set of d-dimensional ε-badly approximable points has packing codimension comparable to ε^d up to multiplicative constant. Any C^2 submanifold of ℝ^d with bounded first and second partial derivatives intersected with the set of d-dimensional ε-badly approximable points has packing dimension strictly less than the dimension of the submanifold. For any matrix A∈ ℝ^m× n satisfying certain Diophantine conditions, the set of points that avoid the ε |q|_∞^−n/m-neighbourhood of the orbit {Aq mod ℤ^m:q ∈ ℤ^n} has packing codimension greater than C_A ε^m for C_A a constant depending on A and m only. \end{itemize}
ER -