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

Abstract

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 𝕋dto ℝ under which the partial sums f ◦ T + f ◦ T2+ ⋯ + f ◦ Tnsatisfy 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 n1/4log n.