Limiting curlicue measures for theta sums

Cellarosi, Francesco

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 466-497.

Nous considérons l'ensemble des courbes {γα, N: α∈(0, 1], N∈ℕ} obtenues en interpolant les valeurs des sommes thêta normalisées N-1/2∑n=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 résultat 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 géométriques de telles courbes, qui présentent des motifs à spirale (curlicues) à différentes échelles. Nous exploitons une procédure de renormalisation construite par le développement de α en fractions continues avec quotients partiels pairs et un théorème de renouvellement pour les dénominateurs de tels développements en fractions continues.

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-1/2∑n=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.

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