An analogue of the nearest integer continued fraction for certain cubic irrationalities

Abstract

Let θ \theta be any irrational and define N e ( θ ) Ne(\theta ) to be that integer such that | θ − N e ( θ ) | > 1 2 |\theta - Ne(\theta )|\; > \frac {1}{2} . Put ρ 0 = θ {\rho _0} = \theta , r 0 = N e ( ρ 0 ) {r_0} = Ne({\rho _0}) , ρ k + 1 = 1 / ( r k − ρ k ) {\rho _{k + 1}} = 1/({r_k} - {\rho _k}) , r k + 1 = N e ( ρ k + 1 ) {r_{k + 1}} = Ne({\rho _{k + 1}}) . Then the r’s here are the partial quotients of the nearest integer continued fraction (NICF) expansion of θ \theta . When D is a positive nonsquare integer, and θ = D \theta = \sqrt D , this expansion is periodic. It can be used to find the regulator of Q ( D ) \mathcal {Q}(\sqrt D ) in less than 75 percent of the time needed by the usual continued fraction algorithm. A geometric interpretation of this algorithm is given and this is used to extend the NICF to a nearest integer analogue of the Voronoi Continued Fraction, which is used to find the regulator of a cubic field F \mathcal {F} with negative discriminant Δ \Delta . This new algorithm (NIVCF) is periodic and can be used to find the regulator of F \mathcal {F} . If I > | Δ | / 148 4 I > \sqrt [4]{{|\Delta |/148}} , the NIVCF algorithm can be used to find any algebraic integer α \alpha of F \mathcal {F} such that N ( α ) = I N(\alpha ) = I . Numerical results suggest that the NIVCF algorithm finds the regulator of F = Q ( D 3 ) \mathcal {F} = \mathcal {Q}(\sqrt [3]{D}) in about 80 percent of the time needed by Voronoi’s algorithm.