Supplementary Materials: Proof of Theorem 2.1

Our argument can be loosely outlined as follows. The bulk of the work is devoted to bounding | ̃̀0(x)|, which takes several steps to accomplish. Taking x as fixed, we determine the choice of the points x̃j that maximizes | ̃̀0(x)| and then bound the maximum using integrals. Since the resulting bound is independent of x̃j , we can exploit symmetry to obtain bounds on | ̃̀k(x)| for k 6= 0. We then sum these bounds over k to obtain a bound on Λ̃N . We begin with the following result, which shows that to bound | ̃̀0(x)| we need consider only grids in which all the points, possibly aside from x̃0, are perturbed by the maximum amount of αh.