Limiting Curlicue Measures for Theta Sums

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Cellarosi, Francesco
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Abstract
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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.
Comment: 36 pages, 3 figures, submitted to Ann. Inst. Henri Poincare' Probab. Stat
Keywords
Mathematics - Dynamical Systems, Mathematics - Number Theory, 60F05, 11F27, 60B10, 11J70, 37E05, 28D05
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