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Banach's problem: a probability-preserving transformation with simple Lebesgue spectrum

Theorem 1.1: there are a $C^\infty$ volume-preserving diffeomorphism $T$ of $\mathbb T^3$ and a real $f\in L^2_0(\mathbb T^3)$ such that $\{f\circ T^n\}_{n\in\mathbb Z}$ is an orthonormal basis of $L^2_0$ ; hence $T$ is ergodic and $U_T$ on $L^2_0$ is unitarily equivalent to multiplication by $w$ on $L^2(S^1)$ . Mixing, zero entropy and zero Lyapunov exponents follow; mixing of all orders uses a separate multiple-mixing manuscript. The manuscript states this is distinct from the historical real-line question attributed to Banach and does not address that formulation; $f$ is only $L^2$ , not smooth.

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Banach's problem: a probability-preserving transformation with simple Lebesgue spectrum · Banach's simple Lebesgue spectrum problem

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VibeMathed record: Banach's problem: a probability-preserving transformation with simple Lebesgue spectrum

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the probability-preserving form of the problem. The proof was not refereed. The Lean challenge ComparatorChallenges/ThreeTorus.json (theorem OAI.ThreeTorus.simple_lebesgue_spectrum, solution module OAI.Dynamics.ThreeTorus.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/144.md. Its statement was read here: a bijection $T$ of $(\mathbb R/\mathbb Z)^3$ , smooth with smooth inverse via local lifts, preserving Haar measure, and a real $f$ in the complex mean-zero $L^2$ whose bilateral iterates are orthonormal with dense span, with $T$ ergodic and a linear isometry onto $L^2$ of the circle conjugating the Koopman operator to multiplication by the first character. This states the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.