The divided cell algorithm and the inhomogeneous Lagrange and Markoff spectra

TL;DR

This paper revisits the divided cell algorithm, linking it to continued fractions and demonstrating how recent computational and theoretical advances enhance its application to inhomogeneous approximation problems.

Contribution

It integrates symbolic computation and spectral theory to improve the utility of the divided cell algorithm for inhomogeneous minima analysis.

Findings

Connected divided cells to continued fractions for easier comparison.

Enhanced understanding of inhomogeneous spectra through modern computational methods.

Unified approach to organize inhomogeneous approximation data.

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.