Littlewood's conjecture

Prove that every pair of real numbers (x,y) satisfies \(\liminf_{q\to\infty} q\,\|qx\|\,\|qy\|=0\), thereby resolving Littlewood's conjecture.

Background

The paper recalls Littlewood's conjecture, which asserts that multiplicative Diophantine approximation in two dimensions cannot remain uniformly bounded away from zero. Although almost every pair satisfies the conjecture and the exceptional counterexample set is known to have Hausdorff dimension zero, the conjecture itself remains unresolved.

References

Also, when $E$ is a cubic field, we do not know whether there exist pairs $(\lambda,\mu)$ of real numbers other than $(1/2,1/2)$, with $0\le\lambda\le1$ and $\lambda+\mu=1$, and such that, for every $ > 0$, there exist arbitrarily large positive integers $Q$ with $$ Q\lambda \, \Vert Q\alp_1 \Vert \ll1 \quad \hbox{and} \quad Q\mu\, \Vert Q\alp_2\Vert \ll . $$

— Refinements of Peck's theorem on simultaneous approximation to algebraic numbers  (2609.29360 - Bugeaud et al., 24 Sep 2026) in Section 5, Additional remarks

Littlewood's conjecture remains far from resolved.

— Winning property of counterexamples to Uniform Littlewood's Conjecture  (2608.24401 - Wu et al., 25 Aug 2026) in Section 1, Subsection 1.1, immediately following the statement of Conjecture (Littlewood)