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arXiv · 0905.1092

Limiting Curlicue Measures for Theta Sums

Francesco Cellarosi
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Abstract

We consider the ensemble of curves $\{γ_{α,N}:α\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(πi n^2α)$, $0\leq N'<N$. We prove the existence of limiting finite-dimensional distributions for such curves as $N\to\infty$, with respect to an absolutely continuous probability measure $μ_R$ 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 $α$ with even partial quotients and a renewal-type limit theorem for the denominators of such continued fraction expansions.

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Francesco Cellarosi. 2009-05-07. Limiting Curlicue Measures for Theta Sums. https://arxiv.org/abs/0905.1092

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