Erdős problem 319 · problems.science

What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $\sum_{n\in A} \frac{\delta n}{n} = 0$ and $\sum_{n\in A'}\frac{\delta n}{n} \neq 0$ for all non-empty $A'\subsetneq A$ .

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319.lean

Retained formal statement5 of 5

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.

FormalConjectures/ErdosProblems/319.leanErdos319.erdos_319.variants.lb9 linesExact file

1∃ o,2  o =o[Filter.atTop] 1 ∧3    ∀ᶠ (N : ℕ) in Filter.atTop,4      (1 - 1 / Real.exp 1 + o N) * ↑N ≤5        sSup6          {x |7            ∃ A,8              ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :9                ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}

SolvedStatement only, no proof

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