$\mathbf S_{\rm erg}$, respectively.

Small subsets of the circle
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Given a family of measures $\mathcal B\subset\mathcal M(\TT)$, it is quite natural in harmonic analysis to try to say something about the *$\mathcal B$-small* subsets of $\TT$, i.e. the sets $D\subset\TT$ that are annihilated by all positive measures $\sigma\in\mathcal B$. By this we mean that for any such measure $\sigma$, one can find a Borel set $\widetilde D$ (possibly depending on $\sigma$) such that $D\subset\widetilde D$ and $\sigma (\widetilde D)=0$. When the family $\mathcal B$ has the form $\mathcal B=\mathcal F_+^{-1}(\mathbf S)$ for some $\mathbf S\subset\ell^\infty (\ZZ_+)$, we call these sets *$\mathbf S$-small*.

To avoid trivialities concerning $\mathcal B$-small sets, the family $\mathcal B$ under consideration should contain nonzero [positive]{} measures, and in fact it is desirable that it should be *hereditary* with respect to absolute continuity; that is, any measure absolutely continuous with respect to some $\sigma\in\mathcal B$ is again in $\mathcal B$. The following simple lemma shows how to achieve this for families of the form $\mathcal F_{+}^{-1}(\mathbf S)$. Let us say that a family $\mathbf S\subset\ell^\infty(\ZZ_+)$ is *translation-invariant* if it is invariant under both the forward and the backward shift on $\ell^\infty(\ZZ_+)$.

\[hereditary\] If $\mathbf S$ is 