Pairwise mixing layouts and self-joinings of weakly mixing transformations

Let T be an invertible weakly mixing transformation of a standard probability space (X, μ). We prove that pairwise mixing along any prescribed finite family of integer sequences implies joint mixing: if every two-coordinate marginal of (T^{d₁(b)}x, …, T^{dₖ(b)}x), for x distributed according to μ, converges to μ², then the full distribution converges to μᵏ. No entropy assumption is needed. This answers Ryzhikov's question on mixing sequences. The proof depends essentially on structural results from OpenAI's multiple-mixing preprint. For weakly mixing rank-one transformations, a tower argument shows that every pairwise-independent finite self-joining is product, answering the corresponding questions in Ryzhikov's problem list. Finally, for invertible ergodic transformations of standard nonatomic spaces, two-fold minimal self-joinings imply minimal self-joinings of all finite orders. This follows from OpenAI's theorem and Ryzhikov's equivalence with the restricted multiple-mixing problem.

GitHub repository: https://github.com/eulogica/mixing-layouts-and-joinings