The spacing of the minima in certain cubic lattices (Q1820816)

From the author's paper: ''Let \({\mathcal K}\) be a cubic field with negative discriminant; let \(\mu,\nu\in {\mathcal K}\); and let \({\mathcal R}\) be a lattice with basis \(\{1,\mu,\nu\}\) such that 1 is a minimum of \({\mathcal R}\). If \(1=\theta_ 1\), \(\theta_ 2,\theta_ 3,...,\theta_ n,..\). is a chain of adjacent minima of \({\mathcal R}\) with \(\theta_{i+1}>\theta_ i\) \((i=1,2,3,...)\), then \(\theta_{n+5}\geq \theta_{n+3}+\theta_ n.\) This result can be used to prove that if p is the period of Voronoi's continued fraction algorithm for finding the fundamental unit \(\epsilon_ 0\) of \({\mathcal K}\), then \(\epsilon_ 0>\tau^{p/2}\), where \(\tau =(1+\sqrt{5})/2\). It is also shown that \(\theta_ n>4^{[(n- 1)/7]}.\)''