Littlewood conjecture

Remarks

The problem is open; Littlewood posed it around 1930. It holds if $\alpha$ is not badly approximable, that is, if $\liminf_n n \norm{n\alpha} = 0$. Indeed $\norm{n\beta} \le 1/2$ gives $n \norm{n\alpha} \norm{n\beta} \le n \norm{n\alpha} / 2$. Rational numbers are not badly approximable, and the same holds with the roles of $\alpha$ and $\beta$ exchanged. By the theory of continued fractions, a number is badly approximable exactly when its partial quotients are bounded, so the open case is a pair of such irrational numbers. Cassels and Swinnerton-Dyer proved it in 1955 when $1, \alpha, \beta$ span a totally real cubic number field. Einsiedler, Katok and Lindenstrauss proved in 2006 that the set of pairs $(\alpha, \beta)$ for which it fails has Hausdorff dimension $0$. Their proof uses the action of the diagonal group on $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$ and the rigidity of measures invariant under it.