| This thesis establishes a number of limit theorems for theta sums. The classical theta sums are of the form n=1Nepin 2a and appear in several problems in Number Theory, Quantum Mechanics, and Optics. More general theta sums of the form n∈Nf ntN e(1/2n²alpha) are also considered. For different values of t, we study the joint distributions of such sums in the complex plane as N → infinity, when alpha is randomly distributed.;For classical theta sums, we establish the existence of the limiting distribution using a renormalization scheme, based on the expansion of alpha into continued fractions with even partial quotients. A renewal-type limit theorem for the sequence of denominators of the rational convergents for alpha---related to the mixing property of a suitably defined special flow---constitutes the core of the proof.;The case of general theta sums is studied in the framework of Homogeneous Dynamics. In this case the existence of the limiting distributions follows from the equidistribution of long, closed horocycles in the unit tangent bundle of a suitably constructed hyperbolic surface. Some properties of these distributions are also discussed. |
