Limit laws for rational continued fractions and value distribution of quantum modular forms
Sandro Bettin, Sary Drappeau
Abstract
We study the limiting distributions of Birkhoff sums of a large class of cost\nfunctions (observables) evaluated along orbits, under the Gauss map, of\nrational numbers in $(0,1]$ ordered by denominators. We show convergence to a\nstable law in a general setting, by proving an estimate with power-saving error\nterm for the associated characteristic function. This extends results of Baladi\nand Vall\\'ee on Gaussian behaviour for costs of moderate growth.\n We apply our result to obtain the limiting distribution of values of several\nkey examples of quantum modular forms. We show that central values of the\nEsterman function ($L$ function of the divisor function twisted by an additive\ncharacter) tend to have a Gaussian distribution, with a large variance. We give\na dynamical, "trace formula free" proof that central modular symbols associated\nwith a holomorphic cusp form for $SL(2,{\\bf Z})$ have a Gaussian distribution.\nWe also recover a result of Vardi on the convergence of Dedekind sums to a\nCauchy law, using dynamical methods.\n