An elementary proof of the canonizing version of Gallai-Witt's theorem

A homothetic mapping (homothety) of the r-dimensional lattice grid N’ is a mapping h: N’ + N’ of the form h(b) = a + db, where a E Nf is a translation vector and d is a positive integer describing a dilatation. A multidimensional version of van der Waerden’s theorem on arithmetic progressions is independently due to Gallai and to Witt (for general references see [S]). It asserts that for every mapping d: IO,..., n l}’ + (0, l}, where n 2 n(t, m) is sufficiently large, there exists a homothety h: N’ + N’ such that d(h(b)) = d(h(c)) for all b, c e (O,..., m l}‘. A canonizing version of this theorem was proved by Deuber, Graham, Promel, and Voigt [ 11. Let Us Cl’ be a linear subspace of the t-dimensional vector space over the rationals. Let d “: N’ + N be a mapping with the property that A,(b) = A,(c) iff b c E U. Of course, A U acts constantly on each coset of U and different cosets get different images. Obviously, A,(h(b)) = A.(h(c)) iff A.(b) = A.(c) for every homothety. Thus, A, induces the same pattern on all homothetic copies of {CL, WI1)‘. A vector b E Q’ is called admissible for SE N’ iff there exists a E Q’ such that the affine line {a + Ib 1 A E Q } intersects S in at least two points. Let