Geometrical Bijections in Discrete Lattices

Abstract

We define uniformly spread sets as point sets in d-dimensional Euclidean space that are wobbling equivalent to the standard lattice ℤ^d. A linear image ϕ(ℤ^d) of ℤ^d is shown to be uniformly spread if and only if det(ϕ) = 1. Explicit geometrical and number-theoretical constructions are given. In 2-dimensional Euclidean space we obtain bounds for the wobbling distance for rotations, shearings and stretchings that are close to optimal. Our methods also allow us to analyse the discrepancy of certain billiards. Finally, we take a look at paradoxical situations and exhibit recursive point sets that are wobbling equivalent, but not recursively so.