Singular Schaeffer-Salem measures of dynamical system origin

TL;DR

This paper investigates a class of measure-preserving dynamical systems with spectral measures exhibiting an extremal decay rate of Fourier coefficients, leading to new constructions of singular distributions of Schaeffer-Salem type on tori and Euclidean spaces.

Contribution

It introduces a novel class of dynamical systems with spectral measures decaying at an extremal rate, expanding the understanding of singular measures of Schaeffer-Salem type.

Findings

Spectral measures decay as |n|^{-1/2+ε} for any ε > 0.

Provides new constructions of singular distributions on T^d and R^d.

Connects dynamical systems with classical singular measure theory.

Abstract

We study a class of dynamical systems given by measure preserving actions of the group $Z^d$ or $R^d$ and generating a set of spectral measures with an extremal rate of the Fourier coefficient decay: $\Hat\sigma(n) = O(|n|^{-1/2+\epsilon})$ for any $\epsilon > 0$. Singular measures with this property are investigated in works due to Wiener and Wintner, Schaeffer, Salem, Ivashev-Musatov, Zygmund et al. Thus, the discovered effect provides a new construction of singular distributions of Schaeffer-Salem type on the torus $T^d$ and in the space $R^d$.