Quantitative weak mixing for typical Salem substitution suspension flows

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.