The spacing of the minima in certain cubic lattices
Hugh C. Williams · Pacific Journal of Mathematics · 1986
Let Jf be a cubic field with negative discriminant; let μ, v e jf and let 01 be a lattice with basis {l,μ,v} such that 1 is a minimum of ^.If 1β 1 , θ 2 , θ 3 ,..., θ n ,... is a chain of adjacent minima of 3t with0 /+1 > 0, (i = 1,2,3,...), then This result can be used to prove that if p is the period of Voronoi's continued fraction algorithm for finding the fundamental unit ε 0 of Jf, then where r = (1 + /5 )/2.It is also shown that 1. Introduction.In order to discuss the problems considered in this paper, it is necessary to give a brief description of the properties of cubic lattices.For a more extensive and more general treatment of these topics we refer the reader to Delone and Faddeev [1].Let f(x) EL Z[X] be a cubic polynomial, irreducible over the rationals M and having a negative discriminant.Let 8 be the real zero of f(x) and denote by JΓ= £(δ) the complex cubic field formed by adjoining 8 to J. If ).For the sake of convenience we will often use the expression a e Se to denote that it is the corresponding point A e S\ that is actually in «£?.Also, if Jδ?= (λ, μ, P), we define a££ (a e Jf) to be the lattice (αλ, αμ, α*>).