About thin arithmetic discrete planes

Arithmetic discrete planes are sets of integer points located within a fixed bounded distance (called thickness) of a Euclidean plane. We focus here on a class of “thin” arithmetic discrete planes, i.e., on a class of arithmetic discrete planes whose thickness is smaller than the usual one, namely the so-called standard one. These thin arithmetic discrete planes have “holes” but we consider a thickness large enough for these holes to be bounded. By applying methods issued from the study of tilings and quasicrystals derived from cut and project schemes, we first consider configurations that occur in thin arithmetic discrete planes. We then discuss substitution rules acting on thin discrete planes, with these geometric rules mapping faces of unit cubes to unions of such faces.