Bugeaud Collection of Conjectures and Open Questions: Rapidly Increasing Sequences Dense Modulo One

formal-conjectures · books · ams-11

Problem 10.6. Find a very rapidly increasing sequence $(m_n)_{n \ge 1}$ of positive
integers such that $(\{\xi m_n\})_{n \ge 1}$ is dense modulo one for every irrational
number $\xi$. Note: Furstenberg's $2^m3^n$ is sublacunary but requires two parameters.

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
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Books/BugeaudDistributionModuloOne/Problem10_6.lean
Local target: Corpus.BooksBugeaudDistributionModuloOneProblem106.problem_10_6_variant_1
Source SHA-256: 4efa4bb14f9b15901bb10763c27010328d892156132ff1da43a6d78bd346745f
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