BOUNDED INTERPOLATIONS BETWEEN LATTICES

It is shown that, given two arbitrary lattices of equal density in the Euclidean space R(n), a bounded quasi-periodic and piecewise affine vector field upsilon on R(n) (a so-called 'modulation field') can be built so that the second lattice is the image of the first one under the map x --> x - upsilon(x). The proof relies on a factorization lemma for matrices with determinant equal to one. Each factor represents a shear-like transformation of R(n) which, in turn, is closely approximated by a periodic set of 'slips' in the lattice.