Discrete Geometry and Covering Problems

Abstract

This thesis explores several problems in discrete geometry, focusing on covering problems. We first go over some well known results, explaining Keith Ball\u27s solution to the symmetric Tarski plank problem, as well as results of Alon and F\ uredi on covering all but vertices of a cube with hyperplanes. The former extensively utilizes techniques from matrix analysis, and the latter applies polynomial method. We state and explore the related problem, asking for the number of parallel hyperplanes required to cover a given discrete set of points in $\mathbb{Z}^{d}$ whose entries are bounded, and prove that there exist sets which are ``difficult\u27\u27 to cover in every dimension for entries whose absolute values are bounded by~1 using a similar polynomial-based approach