Erdős Problem 319

Reference: erdosproblems.com/319

theorem Erdos319.erdos_319 (N : ℕ) : IsGreatest {x : ℕ | ∃ (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ (δ : ℕ → ℤˣ), ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), A.card = x} sorry

What is the size of the largest $A\subseteq\{1, ..., N\}$ 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 $A'\subsetneq A$.

theorem Erdos319.erdos_319.variants.isTheta (N : ℕ) (c : ℕ → ℝ) (h : ∀ (N : ℕ), IsGreatest {x : ℝ | ∃ (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ (δ : ℕ → ℤˣ), ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x} (c N)) : c =Θ[Filter.atTop] sorry

Let $c(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ 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 $A'\subsetneq A$. What is $\Theta(c(N))$?

theorem Erdos319.erdos_319.variants.isBigO (N : ℕ) (c : ℕ → ℝ) (h : ∀ (N : ℕ), IsGreatest {x : ℝ | ∃ (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ (δ : ℕ → ℤˣ), ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x} (c N)) : c =O[Filter.atTop] sorry

Let $c(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ 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 $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N)).

theorem Erdos319.erdos_319.variants.isLittleO (N : ℕ) (c : ℕ → ℝ) (h : ∀ (N : ℕ), IsGreatest {x : ℝ | ∃ (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ (δ : ℕ → ℤˣ), ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x} (c N)) : c =o[Filter.atTop] sorry

Let $c(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ 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 $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = o(g(N)).

theorem Erdos319.erdos_319.variants.lb : ∃ (o : ℕ → ℝ), o =o[Filter.atTop] 1 ∧ ∀ᶠ (N : ℕ) in Filter.atTop, (1 - 1 / Real.exp 1 + o N) * ↑N ≤ sSup {x : ℝ | ∃ (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ (δ : ℕ → ℤˣ), ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}

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.