Furstenberg's `times p, times q` conjectures

formal-conjectures · arxiv · ams-37

**Conjecture 1.3** (the $\times p, \times q$ conjecture): the only atomless Borel probability
measure on $\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/2303.01089
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Arxiv/2303.01089/FurstenbergTimesPTimesQ.lean
Local target: Corpus.Arxiv230301089FurstenbergTimesPTimesQ.conjecture_1_3
Source SHA-256: fba8fb727247fae4f5650843f96e591b02f0ac8c08069b57ce538d11fa007f42
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record