Source labels openChecked July 26, 2026

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

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.

Number theory AMS 11

Mathematical statement

Statement source: Books statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

problem_10_6_variant_1

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem problem_10_6_variant_1 :    ∃ m : ℕ → ℕ,    StrictMono m ∧    IsGenuinelySublacunary m ∧    ∀ ξ : ℝ, Irrational ξ →      Dense (Set.range fun n => (↑(ξ * m n) : AddCircle (1 : ℝ))) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization