Open problem, Arithmetic and number theory, posed 1930

Littlewood conjecture

Open

For every pair of real numbers $(\alpha, \beta) \in \mathbb{R}^2$ , $\liminf_{n \to \infty} n \, \|n\alpha\| \, \|n\beta\| = 0$ , where $\|x\|$ denotes the distance from $x$ to the nearest integer.

Research frontier as of 2026

As of 2026, Littlewood's conjecture remains open. The landmark 2006 theorem of Einsiedler, Katok, and Lindenstrauss proves that the set of exceptions $(\alpha, \beta) \in \mathbb{R}^2$ has Hausdorff dimension $0$ , using measure rigidity for higher-rank diagonal actions on $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3, \mathbb{Z})$ . Subsequent work has verified the conjecture when $\alpha$ lies in specific fractal sets (such as missing-digit Cantor sets) or satisfies explicit continued-fraction growth conditions, and has established quantitative density bounds. However, measure rigidity requires positive entropy, leaving open the possibility of isolated or zero-entropy non-algebraic bounded orbits.

Best known results

Einsiedler, Katok, and Lindenstrauss (2006): the set of pairs $(\alpha, \beta) \in \mathbb{R}^2$ violating Littlewood's conjecture has Hausdorff dimension $0$ .
Cassels and Swinnerton-Dyer (1955): the conjecture holds whenever $1, \alpha, \beta$ span a cubic number field over $\mathbb{Q}$ .

Tools and where they stop

Tool	Achieved	Where it stops
Measure rigidity on homogeneous spaces ( $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3, \mathbb{Z})$ )	Classifies positive-entropy invariant measures under the diagonal action and forces the exceptional set to have Hausdorff dimension $0$	Cannot classify invariant measures of entropy $0$ , so it cannot rule out individual bounded trajectories that carry no positive-entropy measure
Continued fractions and unit groups in number fields	Settles the conjecture when $\alpha$ is not badly approximable or when $(\alpha, \beta)$ lies in a cubic field over $\mathbb{Q}$	Higher dimensions lack a canonical periodic continued-fraction algorithm for general pairs $(\alpha, \beta)$

Open questions

Is every bounded orbit of the diagonal group $A$ in $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3, \mathbb{Z})$ periodic (compact), which would imply Littlewood's conjecture in full?
Does the $p$ -adic Littlewood conjecture $\liminf_{n \to \infty} n \, |n|_p \, \|n\alpha\| = 0$ hold for every real number $\alpha$ and every prime $p$ ?

References

Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss (2006). Invariant measures and the set of exceptions to Littlewood's conjecture · DOI:10.4007/annals.2006.164.513 · arXiv:math/0612721
J. W. S. Cassels, H. P. F. Swinnerton-Dyer (1955). On the product of three homogeneous linear forms and the indefinite ternary quadratic forms · DOI:10.1098/rsta.1955.0010
Manfred Einsiedler, Elon Lindenstrauss (2006). Open problems in dynamical diophantine approximation