Twisted Diophantine Approximation II: Uniform Theory
Abstract
We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted ψ-Dirichlet vectors for a large class of functions ψ, including all except finitely many power functions {T−τ, τ>0}. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted cψ-Dirichlet vectors over c>0, and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of ψ-Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.
Related
- Twisted Diophantine Approximation I: Asymptotic Theory
- 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
BibTeX
@misc{kim2026twisted,
title = {{Twisted Diophantine Approximation II: Uniform Theory}},
author = {Taehyeong Kim and Vasiliy Neckrasov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36412},
eprint = {2609.36412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36412},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisted Diophantine Approximation II: Uniform Theory
AU - Taehyeong Kim
AU - Vasiliy Neckrasov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36412
UR - https://arxiv.org/abs/2609.36412
AB - We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted ψ-Dirichlet vectors for a large class of functions ψ, including all except finitely many power functions {T^−τ, τ>0}. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted cψ-Dirichlet vectors over c>0, and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of ψ-Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.
ER -