Multifractal behavior of polynomial Fourier series

Abstract

We study the spectrum of singularities of a family of Fourier series with polynomial frequencies, in particular we prove that they are multifractal functions. The case of degree two was treated by S. Jaffard in 1996. Higher degrees require completely different ideas essentially because harmonic analysis techniques (Poisson summation) are useless to study the oscillation at most of the points. We introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the Hölder exponent in thin Cantor-like sets.