  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.
