Benjamin Ward

The measure of Badly approximable sets

In classical Diophantine approximation the set of Badly approximable points are those whose rate of approximation, in relation to Dirichlet's Theorem, cannot be improved by an arbitrary small constant. It is a well known result that the lebesgue measure of Badly approximable points is null. The notion of Badly approximable points has natural generalizations in other $n$-dimensional metric spaces. In joint research with Victor Berensevich we prove that in any product space composed of a finite number of bounded separable metric spaces, each equipped with with a $\sigma$-finite doubling Borel regular measure, the product measure of Badly approximable points is null. The proof of this theorem uses standard results from geometric measure theory, including a generalised Lebesgue Density Theorem and Fubini's Theorem.