number theory

Zero-one laws for uniform approximation via Gaussian and Eisenstein integers

arXiv:2607.11178

summary

The paper proves two zero‑one laws for uniform Diophantine approximation of complex numbers by quotients of Gaussian and Eisenstein integers, using homogeneous dynamics and shrinking target problems on SL₂(ℂ) homogeneous spaces.

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 to a shrinking target problem on certain homogeneous spaces of . 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.

44 pages, comments welcome

Topics & keywords

#diophantine approximation#gaussian integers#eisenstein integers#homogeneous dynamics#zero-one lawsuniform approximationshrinking target problemSL_2(C)critical locimeasure estimatesdisjointness
Zero-one laws for uniform approximation via Gaussian and Eisenstein integers · wovepaper