Classical Littlewood conjecture for specific pairs

Determine whether the pairs (√2,√3) and (α,α^{-1}), for an arbitrary real number α, satisfy the classical Littlewood conjecture.

Background

After citing the result that the exceptional set for the classical Littlewood conjecture has Hausdorff dimension zero, the paper emphasizes that this measure-theoretic result does not settle the conjecture for many explicit pairs. In particular, the status of (√2,√3) and of the pair consisting of an arbitrary real number and its reciprocal remains unresolved. Later sections prove the conjecture for certain specially structured continued fractions, including some reciprocal pairs, but not for the full generality stated here.

References

However, despite Proposition \ref{EKLpropintro}, we still do not know whether, for example, $(\sqrt{2},\sqrt{3})$ or $(\alpha,\alpha{-1})$ for an arbitrary $\alpha$ satisfies Littlewoodconjecture.

Littlewoodconjecture:

ℓ(α,β)=0.\ell(\alpha,\beta) = 0.

— Simultaneous approximation to pairs of real numbers  (2608.20028 - Matala-aho, 20 Aug 2026) in Section 1, immediately following Proposition 1