  The note is devoted to multiple mixing, spectrum, rank and self-joinings of
measure-preserving transformations. We recall famous open problems, discuss
related questions and some known results. A hypothetical example of an
automorphism of the class $MSJ(2)\setminus MSJ(3)$ has Lebesgue spectrum,
infinite rank and does not have multiple mixing. If its spectrum is simple,
then we get a solution to the problems of Banach, Rokhlin and del
Junco-Rudolph. The existence of such an example, of course, seems unlikely, but
any facts confirming the impossibility of this amazing situation have not yet
been discovered. They not found even under the condition that its local rank is
positive, which ensures the finite spectral multiplicity.
