Ergodic automorphisms with simple spectrum and rapidly decreasing correlations (Q2440009)

Beginning with the articles of \textit{A. C. Schaeffer} [Am. J. Math. 61, 934--940 (1939; Zbl 0022.35401)] and \textit{R. Salem} [Trans. Am. Math. Soc. 53, 427--439 (1943; Zbl 0060.13709)] several mathematicians studied properties of singular Borel (probability) measures. In particular, they investigated the rate of decay of the Fourier coefficients of such measures. This short article aims to obtain a result in the same spirit in the setting of spectral measures. Let \((X, \mathcal{B}, \mu)\) be a probability space and \(T: X \to X\) be a measure-preserving transformation. For any \(f\in L_2(X, \mathcal{B}, \mu), \) the spectral measure \(\sigma \) of \(f\) is defined via its Fourier coefficients as \[ \Hat{ \sigma}_f (n):=\langle f \circ T^{-n}, f\rangle=\int_X f(T^{-n} x) \overline{f}(x) d \mu (x). \] Let \(\kappa (\sigma) =\inf \{ \alpha \in \mathbb R : \Hat{ \sigma}_f (k) =O(|k|^{\alpha +\epsilon}) \;\forall \epsilon >0 \}. \) The main result states that there exists an invertible measure-preserving transformation \(T\) with simple spectrum such that \(\kappa (\sigma) \leq -\frac12 \) for \(f\) in a dense subset of \(L_2(X, \mathcal{B}, \mu). \) It follows that, for \(f\) in a dense subset of \(L_2(X, \mathcal{B}, \mu), \) one has the rate of decay \( \Hat{ \sigma}_f (k) =O(|k|^{-\frac12 +\epsilon}) \) for all \( \epsilon >0 . \) The author outlines the proof of the theorem by using two separate approaches, which is followed by two conjectures.