Uniform Littlewood conjecture

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

\begin{cases} |p+q\alpha||r+q\beta|<\epsilon/T\\ q\leq T. \end{cases}

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

Equivalent formulations 1

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

The uniform Littlewood conjecture
Let $xi$ and $\zeta$ be real numbers. The uniform Littlewood conjecture asserts that
$\lim_{Q\to\infty} Q \min_{q\in\mathbb{Z},\,0<q\leq Q} \Vert q\xi\Vert\cdot\Vert q\zeta\Vert=0.$
Uniform Littlewood conjecture. For any real numbers $xi,\zeta$ ,
$\lim_{Q\to\infty} Q \min_{q\in\mathbb{Z},\,0<q\leq Q} \Vert q\xi\Vert\cdot\Vert q\zeta\Vert=0.$
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).

Sources & referencesView supporting material

Primary source

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

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Uniform Littlewood conjecture

Equivalent formulations 1

The uniform Littlewood conjecture

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