Bad is null

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.