Renormalisation of curlicues

The recursively spiralling patterns drawn in the complex plane by the values of S-L(tau) = Sigma(L)(n=1) exp(i pi tau n(2)) as L -> infinity with tau fixed in the range 0 <= tau <= 1, depend on the arithmetic of tau. A compendious understanding of the patterns is obtained by iterating an explicit asymptotic renormalisation transformation relating S-L(tau) to a similar sum, magnified by 1/root tau and rotated or reflected, with a smaller number L tau of terms and a new parameter tau(1)(tau). We study some special values of tau, and typical tau. Special values are: (i) tau = 2/L; these are the Gauss sums, whose value (depending on L mod 4) is given exactly by our renormalisation; (ii) rational tau; these have a finite hierarchy of curlicues; if tau = p/q then vertical bar S-L(tau)vertical bar grows linearly if pq is even and repeatedly retraces a finite pattern if pq is odd; (iii) quadratically irrational tau; these are (or are attracted to) fixed points of the map tau(1)(tau); the patterns have an infinite hierarchy of curlicues self-similar under a finite number of scalings, and vertical bar S-L(tau)vertical bar similar to L-1/2; (iv) near-rational tau; these decrease hypergeometrically under the map, with tau(k+1) similar or equal to tau(M)(k); their patterns are self-similar under rapidly increasing scalings and vertical bar S-L(tau)vertical bar similar to LM/(M+1). For typical tau, the fact that the map tau(1)(tau) has a marginally unstable fixed point at tau = 1 causes intermittent sticking of the iterates near tau = 1, and the renormalisation is ineffectual. Our cure is to devise a map with superior ergodic properties, thereby showing that the curlicues are self-similar under a continuum of scalings (the geometric mean magnification per renormalisation being 2.15), and vertical bar S-L(tau)vertical bar similar to L-1/2. The spirallings of S-L(tau) describe the trace of the unitary evolution operator for time T of a rotator with total angular momentum J = h(L(L + 1))(1/2) and Hamiltonian J(z)(2)/2I, in the combined limits of (h) over bar -> 0 (semiclassical), T -> infinity (long time). The intensity of light diffracted by a grating with many slits and detected on a distant screen depends on vertical bar S-L(tau)vertical bar(2). In principle the effects of the curlicues are observable, but experiments are likely to be difficult.