Littlewood conjecture

For $x\in\mathbb R$ let $\|x\|=\min_{m\in\mathbb Z}|x-m|$ be the distance from $x$ to the nearest integer.

Conjecture. For every pair of real numbers $\alpha,\beta$,

$$\liminf_{n\to\infty}\ n\,\|n\alpha\|\,\|n\beta\|=0 ,$$

where $n$ runs over the positive integers.

Equivalently, for every $\varepsilon>0$ there are infinitely many $n\ge1$ with $n\|n\alpha\|\|n\beta\|<\varepsilon$. A solution must prove this for all $(\alpha,\beta)\in\mathbb R^2$.

The statement is easy unless both numbers are badly approximable. A real number $\alpha$ is badly approximable if $\inf_{n\ge1}n\|n\alpha\|>0$, equivalently if the partial quotients of its continued fraction are bounded. If $\alpha$ is not badly approximable, then $\liminf n\|n\alpha\|=0$, and $\|n\beta\|\le1/2$ gives the conjecture. The same holds with $\alpha$ and $\beta$ exchanged. So the open case is a pair $(\alpha,\beta)$ of badly approximable numbers. [7]

Background

Littlewood posed the problem around 1930. It is a question about multiplicative simultaneous approximation. Dirichlet's theorem gives infinitely many $n$ with $\|n\alpha\|\,\|n\beta\|\le n^{-1}$, using the bound $\max(\|n\alpha\|,\|n\beta\|)\le n^{-1/2}$ for each. The conjecture asks for an extra factor tending to $0$.

The problem has a dynamical reformulation. Let $A$ be the group of positive diagonal matrices in $\mathrm{SL}_3(\mathbb R)$ acting on the space $\mathrm{SL}_3(\mathbb R)/\mathrm{SL}_3(\mathbb Z)$ of unimodular lattices in $\mathbb R^3$. The conjecture holds for $(\alpha,\beta)$ when the $A^+$-orbit of the lattice attached to $(\alpha,\beta)$ is unbounded. Margulis conjectured that every bounded $A$-orbit in this space is compact. That conjecture implies Littlewood's conjecture, and it links the problem to measure rigidity for higher-rank diagonal actions [4, 5].