Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limiting curlicue measures for theta sums

Francesco Cellarosi

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We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J. 97 (1999) 127–153] and Jurkat and van Horne [Duke Math. J. 48 (1981) 873–885, Michigan Math. J. 29 (1982) 65–77]. 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.


Nous considérons l’ensemble des courbes {γα, N: α∈(0, 1], N∈ℕ} obtenues en interpolant les valeurs des sommes thêta normalisées N−1/2n=0N'−1exp(πin2α), 0≤N'<N. Nous démontrons l’existence de la limite des distributions fini-dimensionnelles de telles courbes quand N→∞, où α est distribué selon une quelconque mesure de probabilité λ, absolument continue par rapport à la mesure de Lebesgue sur [0, 1]. Notre théorème principal généralise un resultat de Marklof [Duke Math. J. 97 (1999) 127–153] et de Jurkat et van Horne [Duke Math. J. 48 (1981) 873–885, Michigan Math. J. 29 (1982) 65–77]. Notre démonstration se base sur l’analyse des structures géomètriques de telles courbes, qui présentent des motifs à spirale (curlicues) à différentes échelles. Nous exploitons une procédure de renormalisation construite par le developpement de α en fractions continues avec quotients partiels pairs et un théorème de renouvellement pour les dénominateurs de tels developpements en fractions continues.

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Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 466-497.

First available in Project Euclid: 23 March 2011

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Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 11K50: Metric theory of continued fractions [See also 11A55, 11J70] 11J70: Continued fractions and generalizations [See also 11A55, 11K50] 28D05: Measure-preserving transformations 60F99: None of the above, but in this section 60K05: Renewal theory

Theta sums Curlicues Limiting distribution Continued fractions with even partial quotients Renewal-type limit theorems


Cellarosi, Francesco. Limiting curlicue measures for theta sums. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 466--497. doi:10.1214/10-AIHP361.

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