Limiting curlicue measures for theta sums

Francesco Cellarosi

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 466-497
  • ISSN: 0246-0203

Abstract

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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−1/2∑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.

How to cite

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Cellarosi, Francesco. "Limiting curlicue measures for theta sums." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 466-497. <http://eudml.org/doc/241358>.

@article{Cellarosi2011,
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−1/2∑n=0N'−1exp(πin2α), 0≤N'&lt;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.},
author = {Cellarosi, Francesco},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {theta sums; curlicues; limiting distribution; continued fractions with even partial quotients; renewal-type limit theorems; curlicue measures},
language = {eng},
number = {2},
pages = {466-497},
publisher = {Gauthier-Villars},
title = {Limiting curlicue measures for theta sums},
url = {http://eudml.org/doc/241358},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Cellarosi, Francesco
TI - Limiting curlicue measures for theta sums
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 466
EP - 497
AB - 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'&lt;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.
LA - eng
KW - theta sums; curlicues; limiting distribution; continued fractions with even partial quotients; renewal-type limit theorems; curlicue measures
UR - http://eudml.org/doc/241358
ER -

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