On the connectedness of arithmetic hyperplanes

Abstract

Discrete geometry is a geometry specific to computers that studies $Z^d$ structures. It appears naturally in image analysis or 3D printing. Our goal is to find efficient algorithms to characterise these geometric structures and their properties.

We are interested in a fundamental structure of discrete geometry, the arithmetic hyperplanes, and more particularly in their connectedness. Many works have studied a connectedness defined from the neighbourhood by faces and have allowed to observe a percolation phenomenon. These studies have also allowed to decide the connectedness of a plane in an efficient way. We propose an extension of these results in the case of connectivity defined from general neighbourhoods.

Beyond the new concepts that this extension requires, the main contribution of the paper lies in the use of analysis to solve this arithmetic problem and in the design of an algorithm that decides the general connectedness problem. The study of the thickness of connectedness as a function reveals discontinuities at each rational point. However, a much more regular underlying structure appears in the irrational case. Thus, the consideration of irrational vectors allows a simpler approach to the connectedness of arithmetic hyperplanes.