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The analogue of Belinskaya's theorem for measure-preserving flows

Konstantin Slutsky

A measurable real-line set with finite average symmetric-difference measure under unit-interval translations is commensurate with exactly one of the empty set, the whole line, a right half-line or a left half-line. For free ergodic measure-preserving flows, this implies that abstractly isomorphic L¹ full groups determine conjugacy up to a nonzero scalar time change.

Sources for this record

What was already known

Belinskaya’s theorem treated integer actions. Le Maître and Slutsky had a flow criterion assuming positive half-orbits were already commensurate, while group isomorphism was known to imply only the weaker flip Kakutani equivalence.

AI involvement AI-autonomous

Konstantin Slutsky posed the task and started an autonomous Codex GPT-5.5–Claude Opus 4.8 loop. The agents developed the new criterion and application; Slutsky independently verified every step and edited the proofs. François Le Maître commented on an earlier draft.

The central criterion and application emerged from an autonomous two-agent loop after the human posed the task.

Importance Notable

Closes the commensuration gap in a flow rigidity question and strengthens the conclusion from flip Kakutani equivalence to time-change conjugacy.

Limitations

The conclusion concerns free ergodic measure-preserving flows and nonzero scalar time changes. It relies on the earlier human flow criterion; no external or machine proof check is documented.

Technical verification Formal (human)

Slutsky explicitly reports independently checking every mathematical step of the assessed criterion and application. This is a source-reported complete human check.

Formal checking

Independent verification of every mathematical step, followed by final proof editing

Checker
Konstantin Slutsky, paper author
Scope
Section 2 commensuration criterion and Section 3 flow application
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

Slutsky edited the verified proofs and explains how average translation control makes half-orbits commensurate, then invokes the earlier flow conjugacy criterion.

Correction and publication history