Weyl sums, autocorrelations, and a random process.

One of the simplest examples in quantum mechanics, a free particle on the torus, already exhibits complicated autocorrelation functions if the initial wave packet is highly localized. Such autocorrelations can be written in terms of classical number-theoretical sums (Gauss/Weyl sums and generalization thereof). The limiting distribution on the complex plane of Weyl sums can be studied using the theory of continued fractions or homogeneous dynamics. We will focus on the latter approach, and show that the partial sums of a quadratic Weyl sum converge in distribution to a random process on the complex plane. The properties of this process come from the geodesic flow on a hyperbolic manifold. In particular, the anomalous modulus of continuity of sample paths (different from that of a Brownian Motion) is derived from a logarithm law for geodesics by Kleinbock and Margulis. Joint work with J. Marklof.