The following theorem was published in [2].
Theorem. There exists a constant δ > 0 such that for Lebesgue almost every (α, β) ∈ X = [0, 1] × [0, 1], there exists no = no(α, β) such that for any n > no

where the integers pn, qn, rn are provided by the modified Jacobi-Perron algorithm.