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

For an invertible measure-preserving transformation $T$ of a probability space, the Koopman operator $U_Tg=g\circ T$ is unitary on $L^2$ and leaves the constants invariant. Banach's problem, in its probability-preserving form, asks whether some $T$ has simple Lebesgue spectrum on the orthogonal complement $L^2_0$ of the constants: equivalently, whether there is $f\in L^2_0$ whose iterates $\{f\circ T^n\}_{n\in\mathbb Z}$ form an orthonormal basis of $L^2_0$ . Rokhlin (1949) asked for ergodic automorphisms with simple, or at least finite-multiplicity, Lebesgue spectrum. Known constructions gave Lebesgue components, finite-measure flows, simple non-Lebesgue spectrum, or infinite-measure examples, but not this. Does such a probability-preserving transformation exist?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Ergodic theory; spectral theory of measure-preserving maps
Posed by: Stefan Banach (question recorded by Ulam); the probability-space form was asked by V. A. Rokhlin
Year posed: 1949
Years open: 77y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 52 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.

Verification

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.

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

What was actually shown

What the AI did

Verification

Sources

Discussion