Quantitative weak mixing for typical Salem substitution suspension flows

TL;DR

This paper establishes that typical Salem substitution suspension flows exhibit quantitative weak mixing with a rate slightly worse than a power of log log, and proves polynomial decay is impossible for such flows.

Contribution

It demonstrates that for Salem substitution flows, quantitative weak mixing occurs with near-logarithmic rates and provides new decay estimates for Salem Bernoulli convolutions.

Findings

Weak mixing rate is slightly worse than a power of log log.

Polynomial decay of correlations is impossible for these flows.

Provides new decay estimates for Salem Bernoulli convolutions.

Abstract

The paper investigates quantitative weak mixing of Salem substitutions flows. We prove that for a substitution whose substitution matrix is irreducible over the rationals and the dominant eigenvalue is a Salem number, for almost every suspension flow with a piecewise constant roof function, quantitative weak mixing holds with a rate that is slightly worse than a power of $\log\log$. We do not know if this is sharp, but we do show that for any suspension flow of this kind, quantitative weak mixing with a polynomial rate is impossible. Results for specific systems are often much weaker than for ``typical'' or ``generic'' ones. In the Appendix we explain how a minor modification of an argument from Bufetov and Solomyak (2014) yields very weak, but nevertheless quantitative weak mixing estimates of $\log^*$ type for the {\em self-similar} suspension flow over a Salem substitution. Simultaneously this provides first quantitative decay rates for the Fourier transform of Salem Bernoulli convolutions.