Disorder, renormalizability, theta functions and Cornu spirals

The partial sums of the trigonometrical series S = Σ^∞_-∞ e^iπαnp for α small mod 2 lead to distributions of points in the complex plane C composed of Cornu-like spirals. For p ≠ 2 and p > 1 the number N_l of points in the lth spiral is O(l^e) + 1, where e= (2 - p) (p - 1). If p = 2, then N_l = [ 1 2+ (l + 1 2)/α]-[ 1 2+(l - 1 2)/α], [x], denotes the greatest integer in x. Thus N_l increases (decreases) with n if p > 2 (1 < p < 2). We thus have two types of disorderly behavior for p > 1, p ≠ 2 (and p not an integer if α is rational). For p > 2 the number of points per spiral decreases to 1 and the points are distributed in a seemingly random fashion; while for 1 < p < 2 the number of points per spiral increases to + ∞ with l → ∞, and the orientation of the spirals changes in seemingly random fashion with l. There is order if p is an integer: if α is rational the pattern is a pseudo-periodic arrangement of spirals which, depending on α, also may be composed of spirals; if α is a quadratic irrational and p = 2 (i.e. α can be represented by a periodic continued fraction), then the pattern is renormalizable. In the second case the numbers s_l of points between the mid-points (points of inflection) of successive spirals form a Beatty sequence [16]. The proof of renormalizability depends upon Hardy and Littlewood's approximate functional equation for the theta function [6]. Similar behavior is exhibited by n-dimensional generalizations of the sum S above.