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- W2919419678 endingPage "1425" @default.
- W2919419678 startingPage "1377" @default.
- W2919419678 abstract "We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $sum _{ngeqslant 1} tau (n) {rm e}^{2pi i n x}/sqrt {n}$ ( x ∈ Q $xin {mathbb {Q}}$ ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for S L ( 2 , Z ) $SL(2,{mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods." @default.
- W2919419678 created "2019-03-11" @default.
- W2919419678 creator A5019838837 @default.
- W2919419678 creator A5064074045 @default.
- W2919419678 date "2022-09-28" @default.
- W2919419678 modified "2023-09-28" @default.
- W2919419678 title "Limit laws for rational continued fractions and value distribution of quantum modular forms" @default.
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- W2919419678 doi "https://doi.org/10.1112/plms.12485" @default.
- W2919419678 hasPublicationYear "2022" @default.
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