Twisted Diophantine Approximation I: Asymptotic Theory
Abstract
We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous ψ-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.
Related
- Twisted Diophantine approximation: asymptotic and uniform results
- Twisted Diophantine Approximation II: Uniform 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
BibTeX
@misc{kim2026twisted,
title = {{Twisted Diophantine Approximation I: Asymptotic Theory}},
author = {Taehyeong Kim and Vasiliy Neckrasov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36411},
eprint = {2609.36411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36411},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisted Diophantine Approximation I: Asymptotic Theory
AU - Taehyeong Kim
AU - Vasiliy Neckrasov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36411
UR - https://arxiv.org/abs/2609.36411
AB - We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous ψ-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.
ER -