I wondered about those rotated lattices, too. Let L_0 = Z^2 and Let e(t) = exp(2πit) for t in R. So e(t) = e(t') <=> t-t' in Z. L_t = e(t) * L_0 (so that L_t has period π/2) and ask: What is the average number of vertices of L_0 lying in a single cell (Voronoi region) of L_t (or vice versa — it's symmetrical). Possibly this is best asked as a function of the distance R from the origin (common to all L_t's). And asymptotically. So we'd like to know lim (# pts. v=(K,L) with |v| in [R, R + eps) / area(R, R+eps) R—>oo = lim (# pts. v=e(t)*(K,L) with |v| in [R, R + eps) / (π*eps(2*R + eps)) R—>oo where the denominator is the area of the annulus between radii R and R+eps. In case the limit depends on eps > 0, take the additional limit as eps —> 0. —Dan ----- On a similar note, let S be the set of lattice points Z^2 on R^2. Let T be S rotated by angle A about the origin. Is there a bijection f from S to T and constant c such that for all points p in S, |p - f(p)| <= c? -----