FormalConjectures.ErdosProblems.«996»

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import FormalConjecturesUtil

Erdős Problem 996

Reference:

erdosproblems.com/996

[Er49d] Erdös, P. "On the strong law of large numbers." Transactions of the American Mathematical Society 67.1 (1949): 51-56.

[Ma66] Matsuyama, Noboru. "On the strong law of large numbers." Tohoku Mathematical Journal, Second Series 18.3 (1966): 259-269.


open MeasureTheory AddCircle Filter Topology Asymptotics Finset Real


namespace Erdos996


noncomputable def fourierPartial {T : ℝ} [hT : Fact (0 < T)] (f : Lp ℂ 2 (@haarAddCircle T hT))
    (k : ℕ) : AddCircle T → ℂ :=
  fun x => ∑ i ∈ Icc (-k : ℤ) k, fourierCoeff f k • fourier i x

Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?

@[category research open, AMS 42]
theorem declaration uses 'sorry'erdos_996 : answer(sorry) ↔
    ∃ (C : ℝ), 0 < C ∧ ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
    IsLacunary n →
    (fun k => (eLpNorm (fourierPartial f k) 2 (haarAddCircle (T := 1))).toReal) =O[atTop]
    (fun k => 1 / (log (log (log k))) ^ C)
    →
    ∀ᵐ x, Tendsto (fun N => (∑ k ∈ .range N, f (n k • x)) / N) atTop
    (𝓝 (∫ t, f t ∂haarAddCircle)) := by⊢ True ↔
  ∃ C,
    0 < C ∧
      ∀ (f : ↥(Lp ℂ 2 haarAddCircle)) (n : ℕ → ℕ),
        IsLacunary n →
          ((fun k => (eLpNorm (fourierPartial f k) 2 haarAddCircle).toReal) =O[atTop] fun k =>
              1 / log (log (log ↑k)) ^ C) →
            ∀ᵐ (x : AddCircle 1),
              Tendsto (fun N => (∑ k ∈ range N, ↑↑f (n k • x)) / ↑N) atTop
                (𝓝 (∫ (t : AddCircle 1), ↑↑f t ∂haarAddCircle))
  sorryAll goals completed! 🐙

The following theorem is proved in [Ma66].

@[category research solved, AMS 42]
theorem declaration uses 'sorry'erdos_996.variants.log2 : ∀ (C : ℝ), 0.5 < C →
    ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
    IsLacunary n →
    (fun k => (eLpNorm (fourierPartial f k) 2 (haarAddCircle (T := 1))).toReal) =O[atTop]
    (fun k => 1 / (log (log k)) ^ C)
    →
    ∀ᵐ x, Tendsto (fun N => (∑ k ∈ .range N, f (n k • x)) / N) atTop
    (𝓝 (∫ t, f t ∂haarAddCircle)) := by⊢ ∀ (C : ℝ),
  0.5 < C →
    ∀ (f : ↥(Lp ℂ 2 haarAddCircle)) (n : ℕ → ℕ),
      IsLacunary n →
        ((fun k => (eLpNorm (fourierPartial f k) 2 haarAddCircle).toReal) =O[atTop] fun k => 1 / log (log ↑k) ^ C) →
          ∀ᵐ (x : AddCircle 1),
            Tendsto (fun N => (∑ k ∈ range N, ↑↑f (n k • x)) / ↑N) atTop (𝓝 (∫ (t : AddCircle 1), ↑↑f t ∂haarAddCircle))
  sorryAll goals completed! 🐙


end Erdos996