Close Enough: a Dynamical Approach to the Littlewood Conjecture

Diophantine approximation is a branch of number theory that concerns the metric relationship of the rationals and irrationals. Much of the present-day research in the subject approaches problems of approximation via dynamics. We introduce the Littlewood Conjecture, a longstanding problem that was almost completely proven in 2006 by a theorem of Einsiedler, Katok, and Lindenstrauss. After giving an exposition of classical Diophantine approxi- mation and fundamental ergodic theory, we set out a survey of the aforemen- tioned theorem's context, particularly the motivation of the authors' dynam- ical approach, rich connections to linear algebra and hyperbolic geometry, an exploration of measure rigidity, and an interpretation of the paper's main conclusions.