erdos_319.variants.lb

Adenwalla has observed that a lower bound (on the maximum size of $A$) of $$ |A| \geq (1 - \frac{1}{e} + o(1))N $$ follows from the main result of Croot [Cr01]. [Cr01] Croot, III, Ernest S., _On unit fractions with denominators in short intervals_. Acta Arith. (2001), 99-114.

theorem erdos_319.variants.lb : ∃ (o : ℕ → ℝ), (o =o[atTop] (1 : ℕ → ℝ)) ∧
    ∀ᶠ N in atTop, (1 - 1 / rexp 1 + o N) * N ≤ sSup { (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N)
      (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧
        ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) }

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