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For $n\geq 1$ we consider the class JP($n$) of dynamical systems whose everyergodic joining with a Cartesian product of $k$ weakly mixing automorphisms($k\geq n$) can be represented as the independent extension of a joining of thesystem with only $n$ coordinate factors. For $n\geq 2$ we show that, wheneverthe maximal spectral type of a weakly mixing automorphism $T$ is singular withrespect to the convolution of any $n$ continuous measures, i.e. $T$ has theso-called convolution singularity property of order $n$, then $T$ belongs toJP($n-1$). To provide examples of such automorphisms, we exploit spectralsimplicity on symmetric Fock spaces. This also allows us to show that for any$n\geq 2$ the class JP($n$) is essentially larger than JP($n-1$). Moreover, weshow that all members of JP($n$) are disjoint from ergodic automorphismsgenerated by infinitely divisible stationary processes.

Disjointness properties for Cartesian products of weakly mixing systems