We define uniformly spread sets as point sets in d-dimensional
Euclidean space that are wobbling equivalent to the standard lattice
ℤd. A linear image ϕ(ℤd)
of ℤd 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.