A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

Soit $T_{\alpha }$ la rotation sur le cercle d’angle irrationnel $\alpha$, soit ${\left(S_k\right)}_{k\ge 0}$ une marche aléatoire transiente sur $\mathbb{Z}$. Soit $f\in L^2\left(\mu \right)$ et $H\in ]0,1[$, nous étudions la convergence faible de la suite $\frac{1}{n^H}{\sum }_{k=0}^{\left[nt\right]-1}f\circ T_{\alpha }^{S_k},n\ge 1.$