Classical metric Diophantine approximation revisited

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored. 1 Dirichlet, Roth and the metrical theory Diophantine approximation is based on a quantitative analysis of the property that the rational numbers are dense in the real line. Dirichlet’s theorem, a fundamental result of this theory, says that given any real number x and any natural number N , there are integers p and q such that ∣