Homogeneous spectrum, Disjointness of Convolutions, and Mixing Properties of Dynamical Systems

In connection with Rokhlin's question on an automorphism with a homogeneous nonsimple spectrum, we indicate a class of measure-preserving maps $T$ such that $T\times T$ has a homogeneous spectrum of multiplicity 2. The automorphisms in question satisfy the condition $\sigma\ast\sigma\perp \sigma$, where $\sigma$ is the spectral measure of $T$. We also show that there is a mixing automorphism possessing the above properties and their higher order analogs. This work has been published in the pilot issue of Selected Russian Mathematics, Vol.1 (1999), no.1, 13-24.