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Uniform Littlewood conjecture

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

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

A pair is called multiplicatively singular if, for every ϵ>0\epsilon>0, this system has a nonzero integer solution for every sufficiently large TT. Uniform Littlewood conjecture. Every pair (α,β)(\alpha,\beta) is multiplicatively singular; equivalently, for every pair and every ϵ>0\epsilon>0, the system has solutions (p,r,q)∈Z×Z×N(p,r,q)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{N} for all sufficiently large positive TT. The surrounding text states that this question is wide open, although multiplicatively singular pairs have full Lebesgue measure.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The uniform Littlewood conjecture

    Let xixi and ζ\zeta be real numbers. The uniform Littlewood conjecture asserts that

    lim⁡Q→∞Qmin⁡q∈Z, 0<q≤Q∥qξ∥⋅∥qζ∥=0.\lim_{Q\to\infty} Q \min_{q\in\mathbb{Z},\,0<q\leq Q} \Vert q\xi\Vert\cdot\Vert q\zeta\Vert=0.

    Uniform Littlewood conjecture. For any real numbers xi,ζxi,\zeta,

    lim⁡Q→∞Qmin⁡q∈Z, 0<q≤Q∥qξ∥⋅∥qζ∥=0.\lim_{Q\to\infty} Q \min_{q\in\mathbb{Z},\,0<q\leq Q} \Vert q\xi\Vert\cdot\Vert q\zeta\Vert=0.

    The paper states that this conjecture is false and provides counterexamples forming a residual set; consequently, the conjecture is refuted.

    source: Johannes Schleischitz, “Disproof of the uniform Littlewood conjecture”, arXiv:2603.12611 (2026).

References

Primary source

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

diophantine-approximationhomogeneous-dynamicslittlewood-conjectureuniform-approximation
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