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Hausdorff dimension for exactly approximable matrices
Abstract
We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let ψ:N→(0,∞) be a non-increasing approximation function and let τψ denote its lower order. For m,n≥1 and θ∈ℝm, denote Exactm,n(ψ,θ) the set of m× n matrices that are ψ-approximable with shift θ, but are not cψ-approximable for any 0<c<1. It is shown that hdim Exactm,n(ψ,θ)=m(n−1)+(m+n)/(1+τψ), for τψ>n/m. We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces {(p+θ)/q: (p, q)∈ ℤm+1}, {{A∈ Matm, n(R): Aq−p−θ=0}: (p, q)∈ ℤm+n}.
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BibTeX
@misc{he2026hausdorff,
title = {{Hausdorff dimension for exactly approximable matrices}},
author = {Yubin He and Bixuan Li and Baowei Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03442},
eprint = {2610.03442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03442},
}
RIS
TY - UNPB
TI - Hausdorff dimension for exactly approximable matrices
AU - Yubin He
AU - Bixuan Li
AU - Baowei Wang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03442
UR - https://arxiv.org/abs/2610.03442
AB - We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let ψ:N→(0,∞) be a non-increasing approximation function and let τ_ψ denote its lower order. For m,n≥1 and θ∈ℝ^m, denote Exact_m,n(ψ,θ) the set of m× n matrices that are ψ-approximable with shift θ, but are not cψ-approximable for any 0<c<1. It is shown that hdim Exact_m,n(ψ,θ)=m(n−1)+(m+n)/(1+τ_ψ), for τ_ψ>n/m. We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces {(p+θ)/q: (p, q)∈ ℤ^m+1}, {{A∈ Mat_m, n(R): Aq−p−θ=0}: (p, q)∈ ℤ^m+n}.
ER -