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Bugeaud Collection of Conjectures and Open Questions: Rapidly Increasing Sequences Dense Modulo One: problem_10_6_variant_1

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Statement

Problem 10.6. Find a very rapidly increasing sequence (mn)n≥1(m_n)_{n \ge 1} of positive integers such that ({ξmn})n≥1(\{\xi m_n\})_{n \ge 1} is dense modulo one for every irrational number ξ\xi. Note: Furstenberg's 2m3n2^m3^n is sublacunary but requires two parameters.

This is the reading of "very rapidly increasing" as a lower bound on the consecutive ratios, and it is the one that remains open. It is not answered by runs of consecutive integers, since inside a run mn+1−mn=1m_{n+1} - m_n = 1; and by not_isLacunary_of_dense any answer is pinched between a decaying ratio floor and the failure of lacunarity. [Kat16, Cor. 4.9] gives a universally densifying sequence of the multiplicative shape {2n3e}\{2^n 3^e\}, for which the ratio condition reduces to a Diophantine statement about the differences of the exponents ee times log⁡23\log_2 3 modulo one.

Lean statement

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

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Books/BugeaudDistributionModuloOne/Problem10_6.lean.

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