In the search of convergents to [formula omitted

Abstract

Application of continued fractions in high energy physics is well known, especially via the K.A.M. theorem and mostly for quadratic irrationals. Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin’s probability law. Here a combinatorial approach to the search of convergents is presented. We resort to the adjunction ring Z ( 2 3 ) , representing its elements in the irrational basis ρ = 1 + 2 3 + 4 3 .