Khintchine's theorem and related topics in Diophantine approximation

Abstract

An expository overview of Khintchine’s theorem and related developments in metric Diophantine approximation is presented. We begin with a study of the classical theorem and the construction of the Duffin–Schaeffer counterexample that demonstrate the necessity of the monotonicity assumption. Drawing on results of Gallagher, higher-dimensional analogues in which the monotonicity hypothesis can be removed are then explored. Furthermore, we survey recent generalised Duffin–Schaeffer-type counterexamples, illustrating that monotonicity on a support of full upper density remains insufficient. Finally, a metric version of the Littlewood conjecture is discussed, focusing on fibration-type results and on settings in which weaker substitutes for the full Duffin–Schaeffer conjecture suffice.