Numerical Upper Bounds in the Littlewood Conjecture

The famous Littlewood conjecture states that lim infq→∞q∥qα∥∥qβ∥=0 for all α,β∈R, where ∥α∥ denotes the distance of α from the nearest integer. Given ε>0, we introduce a new algorithm based on the simple continued fraction expansions of α and β, to check whether infq∈Z+q∥qα∥∥qβ∥<ε. Using our algorithm we show that infq∈Z+q∥qα∥∥qβ∥<126 for all real numbers α and β.