CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Chacon correlation subsequence prevents mixing

Statement refuted

The normalized Chacon transformation is strongly mixing.

Facts & Assumptions

[F1]

For Chacon, $A = [0, 2 / 9)$ has measure $2 / 9$ and $μ (A \cap T^{- h_{r}} A) \geq 2 / 27$ for all $r \geq 1$ Chacon tower height correlations obstruct mixing. The heights satisfy $h_{r} = (3^{r + 1} - 1) / 2 \to \infty$ Chacon three cut one spacer towers.

[F2]

Assume AC, as in F1 The Axiom of Choice.

Counterexample

Given: The Chacon probability system under AC.

1.1

Use the fixed pair $(A, A)$ from F1. The product of its measures is $(2 / 9)^{2} = 4 / 81$ , while its correlations at the stated heights are at least $2 / 27 = 6 / 81$ . Their difference is therefore at least $6 / 81 - 4 / 81 = 2 / 81 > 0$ for every $r \geq 1$ .

F1F2

2.1

A sequence tending to zero must eventually have absolute value below $1 / 81$ on every subsequence whose indices tend to infinity. The correlations minus their product violate this necessary condition along $h_{r}$ . Thus the strong-mixing conclusion refuted by F1 fails, without any need to assert that the correlations have a subsequential limit. The witness set is fixed, not chosen anew at each height.

F1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Katok–Thouvenot Theorem 5.12 p.697; normalized fixed-level computation (standard reference, not scraped)