Minimal arithmetic thickness connecting discrete planes

Abstract

While connected arithmetic discrete lines are entirely characterized, only partial results exist for the more general case of arithmetic discrete hyperplanes. In the present paper, we focus on the three-dimensional case, that is on arithmetic discrete planes. Thanks to arithmetic reductions on a vector n , we provide algorithms either to determine whether a given arithmetic discrete plane with n as normal vector is connected, or to compute the minimal thickness for which an arithmetic discrete plane with normal vector n is connected.