Rates in the strong invariance principle for ergodic automorphisms of the torus

Let T be an ergodic automorphism of the d-dimensional torus 𝕋d. In the spirit of Le Borgne, we give conditions on the Fourier coefficients of a function f from 𝕋d to ℝ under which the partial sums f ◦ T + f ◦ T2 + ⋯ + f ◦ Tn satisfy a strong invariance principle. Next, reinforcing the condition on the Fourier coefficients in a natural way, we obtain explicit rates of convergence in the strong invariance principle, up to n1/4 log n.