Title: Geometry of numbers and L^p best approximations
Abstract: A theorem of Davenport and Schmidt treats the question of when Dirichlet’s theorem on the rational approximation of irrationals can be improved and if so, by how much. We consider a generalization of this question in the context of the geometry of numbers in R^2, with the sup-norm replaced by the L^p norm for p \geq 1. We obtain sharp bounds for how much improvement is possible under various conditions. The proofs use semi-regular continued fractions that are characterized by a certain best approximation property determined by the norm. We conclude by extending some of the results to the non-convex case 0<p<1. This is joint work with W. Duke, Z. Hacking, and A. Woodall.
