  In 1949 V.A. Rokhlin introduced into ergodic theory the k-fold mixing and
puzzled the mathematical community with the problem of the mismatch of these
invariants. Here's what Rokhlin wrote:
  "The proposed work arose from the author's attempts to solve the well-known
spectral problem of the theory of dynamical systems: are there any metrically
different dynamical systems with the same continuous (in particular, Lebesgue)
spectrum? Using new metric invariants introduced for this purpose, the author
tried to find among the ergodic automorphisms of compact commutative groups
automorphisms of various metric types. It turned out, however, that for all the
indicated automorphisms the new invariants are exactly the same."
  We discuss this long time open problem.
