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

TL;DR

This paper establishes conditions on Fourier coefficients of functions for ergodic automorphisms of the torus that ensure the Birkhoff sums satisfy a strong invariance principle, with explicit convergence rates.

Contribution

It provides new Fourier coefficient conditions that guarantee a strong invariance principle for ergodic automorphisms of the torus, including explicit convergence rates.

Findings

Conditions on Fourier coefficients ensure strong invariance principle.

Explicit rates of convergence are derived.

Applicable to ergodic automorphisms of the torus.

Abstract

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.