Limiting Curlicue Measures for Theta Sums
| dc.creator | Cellarosi, Francesco | |
| dc.date | 2009-05-07 | |
| dc.date.accessioned | 2026-07-07T13:12:43Z | |
| dc.date.available | 2026-07-07T13:12:43Z | |
| dc.description | 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. | |
| dc.description | 36 pages, 3 figures, submitted to Ann. Inst. Henri Poincare' Probab. Stat | |
| dc.identifier | https://arxiv.org/abs/0905.1092 | |
| dc.identifier | http://arxiv.org/abs/0905.1092 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229659 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 60F05, 11F27, 60B10, 11J70, 37E05, 28D05 | |
| dc.title | Limiting Curlicue Measures for Theta Sums | |
| dc.type | text |