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

References:

• [Bos94] Boshernitzan, Michael D. "Density modulo 1 of dilations of sublacunary sequences." Advances in Mathematics 108.1 (1994): 104-117.
• [Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.
• [Fur67] Furstenberg, H. "Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation". Math. Systems Theory 1, 1–49 (1967).
• [Mat80] de Mathan, Bernard. "Numbers contravening a condition in density modulo 1." Acta Mathematica Hungarica 36.3-4 (1980): 237-241.
• [Pol79] Pollington, Andrew Douglas. "On the density of sequence $\{n_ {k}\xi\} $." Illinois Journal of Mathematics 23.4 (1979): 511-515.

theorem Bugeaud06.pollington_de_mathan (m : ℕ → ℕ) (hm : ∀ (n : ℕ), 0 < m n) (hlac : IsLacunary m) : dimH {ξ : ℝ | ¬Dense (Set.range fun (n : ℕ) => ↑(ξ * ↑(m n)))} = 1

The Pollington–de Mathan theorem [Pol79][Mat80]. For every lacunary sequence $(m_n)_{n \ge 1}$ of positive integers, the set of real numbers $\xi$ for which $(\{\xi m_n\})_{n \ge 1}$ is not dense modulo one has full Hausdorff dimension.

theorem Bugeaud06.problem_lacunary_not_dense_of_pollington_de_mathan (h : ∀ (m : ℕ → ℕ), (∀ (n : ℕ), 0 < m n) → IsLacunary m → dimH {ξ : ℝ | ¬Dense (Set.range fun (n : ℕ) => ↑(ξ * ↑(m n)))} = 1) : ∃ (m : ℕ → ℕ), (∀ (n : ℕ), 0 < m n) ∧ IsLacunary m ∧ ¬∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun (n : ℕ) => ↑(ξ * ↑(m n)))

The Pollington–de Mathan theorem implies that a lacunary sequence cannot answer Problem 10.6.

theorem Bugeaud06.furstenberg_two_three (ξ : ℝ) (hξ : Irrational ξ) : Dense {x : AddCircle 1 | ∃ (m : ℕ) (n : ℕ), 0 < m ∧ 0 < n ∧ x = ↑(ξ * ↑(2 ^ m * 3 ^ n))}

Furstenberg's theorem [Fur67] (the $\times 2, \times 3$ case). For every irrational number $\xi$, the two-parameter family $(\{\xi \, 2^m 3^n\})_{m, n \ge 1}$ is dense modulo one.

theorem Bugeaud06.boshernitzan (r : ℕ → ℝ) (hr : ∀ (n : ℕ), 0 < r n) (hunb : ¬BddAbove (Set.range r)) (hsub : Filter.Tendsto (fun (n : ℕ) => r (n + 1) / r n) Filter.atTop (nhds 1)) : dimH {ξ : ℝ | ¬Dense (Set.range fun (n : ℕ) => ↑(ξ * r n))} = 0

Boshernitzan's theorem [Bos94]. Given a real sublacunary sequence $r$, the set of real numbers $\xi$ for which $(\{\xi r_n\})_{n \ge 1}$ is not dense modulo one has Hausdorff dimension zero.

noncomputable def Bugeaud06.mSeq : ℕ → ℕ

The sequence defined by $m_0 = 2$ and $m_{n+1} = \lceil m_n (1 + 1/\log n) \rceil$.

Equations

• Bugeaud06.mSeq 0 = 2
• Bugeaud06.mSeq n.succ = ⌈↑(Bugeaud06.mSeq n) * (1 + 1 / Real.log ↑n)⌉₊

def Bugeaud06.IsGenuinelySublacunary (m : ℕ → ℕ) : Prop

The sequence $m$ eventually grows at least geometrically with a logarithmic correction.

Equations

• Bugeaud06.IsGenuinelySublacunary m = ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, 1 + c / Real.log ↑n ≤ ↑(m (n + 1)) / ↑(m n)

theorem Bugeaud06.example_isGenuineSublacunary : IsGenuinelySublacunary mSeq

The sequence mSeq, given by $m_{n+1} = \lceil m_n (1 + 1/\log n) \rceil$, is genuinely sublacunary: taking $c = 1$, we have $m_{n+1}/m_n \ge 1 + 1/\log n$ because $\lceil m_n (1 + 1/\log n) \rceil \ge m_n (1 + 1/\log n)$.

def Bugeaud06.HasIntermediateGrowth (α : ℝ) (m : ℕ → ℕ) : Prop

The sequence $m$ eventually grows at least as fast as $\exp(n^{\alpha})$, i.e., super-exponential growth when $\alpha > 1$, and stretched-exponential when $0 < \alpha < 1$.

Equations

• Bugeaud06.HasIntermediateGrowth α m = ∀ᶠ (n : ℕ) in Filter.atTop, Real.exp (↑n ^ α) ≤ ↑(m n)

theorem Bugeaud06.example_hasIntermediateGrowth (α : ℝ) (hα₀ : 0 < α) (hα₁ : α < 1) : HasIntermediateGrowth α mSeq

mSeq has intermediate (subexponential but super-polynomial) growth: for every 0 < α < 1 its terms eventually dominate $\exp(n^\alpha)$.

theorem Bugeaud06.problem_10_6_variant_1 : ∃ (m : ℕ → ℕ), StrictMono m ∧ IsGenuinelySublacunary m ∧ ∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun (n : ℕ) => ↑(ξ * ↑(m n)))

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.

theorem Bugeaud06.problem_10_6_variant_2 : ∃ (m : ℕ → ℕ), StrictMono m ∧ (∃ (α : ℝ), 0 < α ∧ α < 1 ∧ HasIntermediateGrowth α m) ∧ ∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun (n : ℕ) => ↑(ξ * ↑(m n)))

Problem 10.6, intermediate-growth variant.