Uniform Diophantine approximation with restrictions via total density of collections of subspaces

TL;DR

This paper develops a framework to prove the uncountability and density of singular Diophantine systems under various restrictions, extending classical results to new settings like inhomogeneous and prime denominator approximations.

Contribution

It introduces a method to establish total density for diverse Diophantine systems, generalizing Khintchine's classical results to broader contexts.

Findings

Proves uncountability and density of singular systems in various Diophantine approximation settings.

Establishes total density for inhomogeneous approximation with fixed translation.

Extends results to approximation with prime denominators and other restrictions.

Abstract

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.