The divided cell algorithm and the inhomogeneous Lagrange and Markoff spectra

Abstract

The divided cell algorithm was introduced by Delone in 1947 to calculate the inhomogeneous minima of binary quadratic forms and developed further by E. S. Barnes and H. P. F. Swinnerton-Dyer in the 1950s. We show how advances of the past fifty years in both symbolic computation and our understanding of homogeneous spectra can be combined to make divided cells more useful for organizing information about inhomogeneous approximation problems. A crucial part of our analysis relies on work of Jane Pitman, who related the divided cell algorithm to the regular continued fraction algorithm. In particular, the relation to continued fractions allows two divided cells for the same problem to be compared without stepping through the chain of divided cells connecting them.