Rates in the strong invariance principle for ergodic automorphisms of   the torus
  Let T be an ergodic automorphism of the d-dimensional torus. In the spirit of Le Borgne, we give conditions on the Fourier coeffi cients of a real valued function f under which the Birkhoff sums satis fy a strong invariance principle. Next, reinforcing the condition on the Fourier coeffi cients in a natural way, we obtain explicit rates of convergence in the strong invariance principle. 
J\'er\^ome Dedecker (MAP5), Florence Merlev\`ede (LAMA), Francoise   Pene (LM)
math.DS math.PR