Hausdorff Discretizations of Algebraic Sets and Diophantine Sets

Abstract

This paper is a continuation of our works in which we study the properties of a new framework for discretization of closed sets based on Hausdorff metric. Let F be a nonempty closed subset of Rn; S ⊆ Zn is a Hausdorff discretization of F if it minimizes the Hausdorff distance to F. We study the properties of Hausdorff discretizations of algebraic sets. Actually we give some decidable and undecidable properties concerning Hausdorff discretizations of algebraic sets and we prove that some Hausdorff discretizations of algebraic sets are diophantine sets. We refine the last results for algebraic curves and more precisely for straight lines.