Twisted Diophantine approximation: asymptotic and uniform results
Abstract
Fix an m× n matrix A; twisted Diophantine approximation studies how well we can approximate the affine form A bf q−bf b by an integer m-vector, depending on the choice of m-vector bf b. We report on two companion papers in twisted Diophantine approximation, covering two setups: asymptotic and uniform. The asymptotic paper gives an exact Khintchine-type Lebesgue zero--one law, a null--conull criterion for badly approximable shifts, and Hausdorff-measure tests yielding exact power-law dimensions under a critical divergence condition. The uniform paper computes Hausdorff dimensions of sets of uniformly approximable vectors. It also gives Hausdorff zero--full laws, uniform-exponent spectra, and a sharp escape-of-mass bound. In both cases the results are established for arbitrary approximation functions from suitable regularity classes, and the arguments use a homogeneous lattice count involving all successive minima to record clustering in every independent direction.
Related
- Twisted Diophantine Approximation I: Asymptotic Theory
- Twisted Diophantine Approximation II: Uniform Theory
- Galois groups of twisted reciprocal polynomials: a uniform asymptotic
- Metric Diophantine approximation on planar and Veronese curves
- Extreme value laws for intrinsic Diophantine approximation on spheres
BibTeX
@misc{kim2026twisted,
title = {{Twisted Diophantine approximation: asymptotic and uniform results}},
author = {Taehyeong Kim and Vasiliy Neckrasov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05824},
eprint = {2611.05824},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05824},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisted Diophantine approximation: asymptotic and uniform results
AU - Taehyeong Kim
AU - Vasiliy Neckrasov
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05824
UR - https://arxiv.org/abs/2611.05824
AB - Fix an m× n matrix A; twisted Diophantine approximation studies how well we can approximate the affine form A bf q−bf b by an integer m-vector, depending on the choice of m-vector bf b. We report on two companion papers in twisted Diophantine approximation, covering two setups: asymptotic and uniform. The asymptotic paper gives an exact Khintchine-type Lebesgue zero--one law, a null--conull criterion for badly approximable shifts, and Hausdorff-measure tests yielding exact power-law dimensions under a critical divergence condition. The uniform paper computes Hausdorff dimensions of sets of uniformly approximable vectors. It also gives Hausdorff zero--full laws, uniform-exponent spectra, and a sharp escape-of-mass bound. In both cases the results are established for arbitrary approximation functions from suitable regularity classes, and the arguments use a homogeneous lattice count involving all successive minima to record clustering in every independent direction.
ER -