Uniform Littlewood’s conjecture

The uniform conjecture, introduced by Bandi, Fregoli, and Kleinbock, asserts that every pair of real numbers satisfies the relevant uniform multiplicative approximation condition. The newer work claims this is false already for pairs $(\xi,\zeta)\in\mathbb{R}^{2}$; this does not disprove the classical, non-uniform Littlewood conjecture.

Known results

• Bandi, Fregoli, and Kleinbock (2025) proved that multiplicatively singular pairs have full Lebesgue measure, while the universal uniform assertion was then described as open.

2026 claimed disproof and largeness result

A 2026 preprint claims a counterexample set $\Theta$ satisfying a positive uniform lower bound is dense $G_\delta$, has full packing dimension, and obeys $\Theta+\Theta=\mathbb{R}^{2}$; it also reports Hausdorff dimension at least $2/3$. A second 2026 preprint establishes a related negative result for the fully inhomogeneous formulation. These are preprint claims and remain unverified.

Current status (as of August 2026): The uniform homogeneous conjecture is claimed false, with a dense and dimensionally large counterexample set, but the claim is unverified; the classical Littlewood conjecture remains open.