Bad is null

TL;DR

This paper introduces a general framework for badly approximable points in metric spaces with measures, proving their measure is zero under mild conditions and applying it to various Diophantine and dynamical systems contexts.

Contribution

It develops a unified approach to badly approximable points in metric spaces and demonstrates their measure-zero property across multiple mathematical settings.

Findings

Measure of badly approximable points is zero under mild assumptions

Framework applies to Diophantine approximation on manifolds and fractals

Extends to weighted and $S$-arithmetic Diophantine approximations

Abstract

In this paper we develop a general framework of badly approximable points in a metric space $X$ equipped with a $\sigma$-finite doubling Borel regular measure $\mu$. We establish that under mild assumptions the $\mu$-measure of the set of badly approximable points is always zero. The framework can be applied to a variety of settings in Diophantine approximation and dynamical systems, which we also consider, including weighted and $S$-arithmetic Diophantine approximations, Diophantine approximation on manifolds and intrinsic approximations on fractals.