MEAN LI-YORKE CHAOS ALONG POLYNOMIALS OF SEVERAL VARIABLES AND PRIME NUMBERS

The paper proves the two target statements (Theorems 1.1–1.2) using a Pinsker-disintegration and subsequence argument together with a polynomial “thinness” lemma and a prime W-trick, then extracts a Cantor scrambled set; all of these components appear explicitly in the uploaded PDF. The candidate solution arrives at the same conclusions via a closely related but not identical route: it emphasizes L2 convergence to characteristic factors, formulates the relatively independent self-joining λ and its diagonal-null property, and uses Mycielski’s theorem explicitly. Substantively, both establish limsup>0 and liminf=0 along arbitrary Følner sequences for polynomials, and the analogous results along primes. Minor presentation gaps in the paper (e.g., an implicit use of non-atomicity to claim λ(ΔX)=0 and an imprecise final Mycielski extraction sentence) do not alter correctness. Hence both are correct, with different proof presentations. For key places in the paper: the statements of Theorems 1.1–1.2 are on the first page of Section 1, and the proofs use Proposition 3.3 (thinness), Theorem 3.1 (Leibman’s L2 convergence), Proposition 3.2/Lemma 3.5 (disintegration), Theorem 4.2 (Pinsker for λ), and Section 5 for primes via W-trick; see the highlighted excerpts .