Littlewood Conjecture

The conjecture states that for every pair of real numbers $\alpha,\beta$ one has $\liminf_{n\to\infty}n\,\|n\alpha\|\,\|n\beta\|=0$, where $\|x\|$ denotes the distance from $x$ to the nearest integer. Informally, no two reals can both resist rational approximation along a common sequence of denominators. The problem is traditionally attributed to Littlewood around 1930, although no securely dated written proposal is known.

Cassels and Swinnerton-Dyer reformulated the question in terms of products of three homogeneous linear forms, opening the connection with what became the theory of diagonal flows on spaces of lattices [CasselsSwinnertonDyer1955]. That dynamical route led to the strongest known result: Einsiedler, Katok, and Lindenstrauss proved that the set of pairs $(\alpha,\beta)$ violating the conjecture has Hausdorff dimension zero [EinsiedlerKatokLindenstrauss2006]. Since the statement holds trivially unless both numbers are badly approximable, any counterexample already lies in a thin set.

A set of dimension zero can still be nonempty, and even simple explicit pairs of quadratic irrationals remain unsettled. A resolution requires showing the exceptional set is empty, or producing a pair for which the product stays bounded away from zero; the conjecture remains a central open problem of Diophantine approximation [Moshchevitin2012Open].