our main concerns are really the weak and strong mixing properties.

At this point, we should recall the definitions. A measure-preserving transformation $T:(X,\mathfrak B,\mu)\to(X,\mathfrak B,\mu)$ is *weakly mixing* (with respect to $\mu$) if $$\frac{1}{N}\sum_{n=0}^{N-1} \vert \mu(A\cap T^{-n}(B))-\mu (A)\mu(B)\vert\xrightarrow{N\to\infty} 0$$ for any measurable sets $A,B\subset X$; and $T$ is *strongly mixing* if $$\mu(A\cap T^{-n}(B))\xrightarrow{n\to\infty} \mu(A)\mu(B)$$ for any $A,B\in\mathfrak B$. (Ergodicity can be defined exactly as weak mixing, but removing the absolute value in the Cesàro mean).

According to the “spectral viewpoint" on ergodic theory, weakly mixing transformations are closely related to *continuous* measures on the circle $\TT$, and strongly mixing transformations are related to *Rajchman* measures, i.e. measures whose Fourier coefficients vanish at infinity. Without going into any detail at this point, we just recall that, by a classical result of Wiener (see [@Ktz]), continuous measures on $\TT$ are characterized by the behaviour of their Fourier coefficients: a measure $\sigma$ is continuous if and only if $$\frac{1}{N}\sum_{n=0}^{N-1} \vert \widehat\sigma (n)\vert\xrightarrow{N\to\infty} 0\, .$$ Wiener’s lemma is usually stated with symmetric Cesàro means, but this turns out to be equivalent. Likewise, by the so-called *Rajchman’s lemma*, a measure $\sigma$ is Rajchman if and only if $\widehat\sigma (n)\to 0$ as $n\to +\infty$ (that is, 