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Zero-one laws for uniform approximation via Gaussian and Eisenstein integers
Abstract
We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing it to a shrinking target problem on certain homogeneous spaces of SL2(ℂ). The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.
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BibTeX
@misc{pfitscher2026zeroone,
title = {{Zero-one laws for uniform approximation via Gaussian and Eisenstein integers}},
author = {René Pfitscher and Anurag Rao and Shucheng Yu and Han Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11178},
eprint = {2607.11178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11178},
}
RIS
TY - UNPB
TI - Zero-one laws for uniform approximation via Gaussian and Eisenstein integers
AU - René Pfitscher
AU - Anurag Rao
AU - Shucheng Yu
AU - Han Zhang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.11178
UR - https://arxiv.org/abs/2607.11178
AB - We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing it to a shrinking target problem on certain homogeneous spaces of SL_2(ℂ). The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.
ER -