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 $A'\subsetneq A$.

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}