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Littlewood's conjecture

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Let (α,β)(\alpha,\beta) be a pair of real numbers, and let ϵ>0\epsilon>0. The system

{∣p+qα∣∣r+qβ∣<ϵ/Tq≤T\begin{cases} |p+q\alpha||r+q\beta|<\epsilon/T\\ q\leq T \end{cases}

uses solutions (p,r,q)∈Z×Z×N(p,r,q)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{N} for an unbounded set of T>0T>0. Littlewood's conjecture. For every pair (α,β)(\alpha,\beta) and every ϵ>0\epsilon>0, there is an unbounded set of T>0T>0 for which the system has a solution. Almost every pair satisfies this property, and the set of multiplicatively badly approximable pairs has zero Hausdorff dimension, but the conjecture remains unresolved.

References

Primary source

Dmitry Kleinbock and Chengyang Wu, “Simultaneously bounded and dense orbits for commuting Cartan actions”, arXiv:2509.05272 (2025).

diophantine-approximationhomogeneous-dynamics
Posted

Progress summary

Refreshed
Claimed progress

A new paper proves the conjecture for a broad special class, but the conjecture for all pairs remains open.

Littlewood posed the conjecture in 1909: every pair of real numbers should admit arbitrarily strong simultaneous multiplicative rational approximations. The general assertion is unresolved.

Known results

  • Cassels and Swinnerton-Dyer (1955): it holds when the two numbers lie in the same cubic field.
  • Pollington and Velani (2000): for each badly approximable α\alpha, a full-Hausdorff-dimension set of badly approximable β\beta satisfies a strong form.
  • Einsiedler, Katok, and Lindenstrauss (2006): the exceptional set has Hausdorff dimension 00.
  • Several later papers prove it for restricted classes defined through continued fractions or recurrence conditions.

2026 developments

An August preprint proves the classical conjecture for a broad stated class using continued-fraction constructions, while explicitly leaving the full conjecture open. A separate March preprint disproves a stronger uniform variant, not Littlewood’s original conjecture.

Current status (as of August 2026): Restricted classes are settled and the exceptional set is extremely small, but Littlewood’s conjecture for every pair remains open.

Sources

Solutions 0

No solutions have been posted yet.