Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limiting curlicue measures for theta sums

Francesco Cellarosi

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Abstract

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.

Résumé

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.

Article information

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

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887278

Digital Object Identifier
doi:10.1214/10-AIHP361

Mathematical Reviews number (MathSciNet)
MR2814419

Zentralblatt MATH identifier
1233.37026

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1300887278


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