Inhomogeneous Diophantine Approximation via the Divided Cell Algorithm

An earlier talk in this seminar (October 22, 2009) described the Divided Cell Algorithm as applied to inhomogeneous approximation problems associated with a particular quadratic form of discriminant 165. Since then, work on that example has been completed and other examples have been studied. This led to greater fluency in working with Divided Cells, allowing the use of this method in a broader class of problems. The present talk depends on the earlier one only by highlighting what has been learned since then.

We begin with an overview of continued fractions to emphasize similarities between homogeneous and inhomogeneous problems. This leads to the definition: given the integer lattice and a pair of intersecting lines in the plane, a Divided Cell is a fundamental parallelogram of the lattice that has one vertex in each quadrant of the complement of the lines. Given one such cell, there is a natural construction leading a chain of such cells and the vertices in each quadrant run through the vertices of the convex hull of the lattice points in the quadrant. Existence of Divided Cells is proved using Continued Fractions.

With an algorithm in hand, we investigate whether it can be used to construct approximation problems that will have prescribed properties.