Limiting Curlicue Measures for Theta Sums

TL;DR

This paper establishes the existence of limiting distributions for curves derived from normalized theta sums, revealing spiral patterns and employing continued fraction analysis for proof.

Contribution

It generalizes previous results by proving the convergence of finite-dimensional distributions for theta sum curves using a novel renormalization approach.

Findings

Existence of limiting finite-dimensional distributions for theta sum curves.

Identification of spiral-like patterns (curlicues) at multiple scales.

Application of continued fraction expansion and renewal theory in the proof.

Abstract

We consider the ensemble of curves $\{\gamma_{\alpha,N}:\alpha\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(\pi i n^2\alpha)$, $0\leq N'<N$. 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.