The problems linked with the so-called 'small divisors', i.e. e. the near resonances of frequencies in a quasiperiodic motion, have been known and studied since the 19th century. Poincaré was not certain about the possibility of overcoming them and only beginning with Siegel's Theorem in 1942 a satisfactory theory of stability of quasiperiodic motions has been developed. In a few cases Yoccoz proved that it's even possible to introduce a purely arithmetical function, built using the continued fraction development of the rotation number, which allows to compute the size of the stability domain of the orbits. More precisely, in 1988 Yoccoz proved that the size of the stability domain (Siegel disk) around an irrationally indifferent fixed point in the complex plane is given by a purely arithmetic function--called Brjuno's function--up to a more regular (L^\infty) correction.

Quite surprisingly, very similar functions are used to investigate the convergence of trigonometric sums involving the divisor function, as discovered by Wilton almost a century ago, and even in the study of differentiability properties of integrals of modular forms.

The Hölder interpolation conjecture (aka Marmi-Moussa-Yoccoz conjecture) states that for quadratic polynomials the correction to be applied to the Brjuno function to obtain the conformal radius of the Siegel disk is in fact 1/2-Hölder continuous. An analogous version of the conjecture stands also for other dynamical systems, including the standard family.

1/2-Hölder continuity seems to be the relevant regularity for these problems also since it measures the difference between formulations of the arithmetical function corresponding to different continued fraction algorithms (Gauss, nearest integer, by-excess,...).

The talk will be based on recent work in collaboration with Seul Bee Lee.