Limiting Curlicue Measures for Theta Sums

Abstract

We consider the ensemble of curves $\{{\gamma }_{\alpha ,N}:\alpha \in (0,1],N\in \mathbb{N}\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}{\sum }_{n=0}^{N^{\prime }-1}\exp (\pi i{n}^2\alpha )$, $0\le N^{\prime }<N$. We prove the existence of limiting finite-dimensional distributions for such curves as $N\to \infty$, with respect to an absolutely continuous probability measure ${\mu }_R$ on $(0,1]$. Our Main Theorem generalizes a result by Marklof and Jurkat and van Horne. Our proof relies on the analysis of the geometric structure of such curves, which exhibit spiral-like patterns (curlicues) at different scales. We exploit a renormalization procedure constructed by means of the continued fraction expansion of $\alpha$ with even partial quotients and a renewal-type limit theorem for the denominators of such continued fraction expansions.

36 pages, 3 figures, submitted to Ann. Inst. Henri Poincare' Probab. Stat

References in corpus (1)

• Symbolic coding for linear trajectories in the regular octagon