1. FormalConjectures
  2. ErdosProblems
  3. «319»
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Erdős Problem 319

Reference: erdosproblems.com/319

open Filter open scoped Topology Finset Real namespace Erdos319

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 $A'\subsetneq A$.

@[category research open, AMS 5] theorem declaration uses 'sorry'erdos_319 (N : ℕ) : IsGreatest { #A | (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) } answer(sorry) := N:ℕ⊢ IsGreatest {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), #A = x} sorry All goals completed! 🐙

Let $c(N)$ be 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 $A'\subsetneq A$. What is $\Theta(c(N))$?

@[category research open, AMS 5] theorem declaration uses 'sorry'erdos_319.variants.isTheta (N : ℕ) (c : ℕ → ℝ) (h : ∀ N, IsGreatest { (#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) } (c N)) : c =Θ[atTop] (answer(sorry) : ℕ → ℝ) := N:ℕc:ℕ → ℝh:∀ (N : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑(#A) = x} (c N)⊢ c =Θ[atTop] sorry All goals completed! 🐙

Let $c(N)$ be 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 $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N)).

@[category research open, AMS 5] theorem declaration uses 'sorry'erdos_319.variants.isBigO (N : ℕ) (c : ℕ → ℝ) (h : ∀ N, IsGreatest { (#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) } (c N)) : c =O[atTop] (answer(sorry) : ℕ → ℝ) := N:ℕc:ℕ → ℝh:∀ (N : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑(#A) = x} (c N)⊢ c =O[atTop] sorry All goals completed! 🐙

Let $c(N)$ be 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 $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = o(g(N)).

@[category research open, AMS 5] theorem declaration uses 'sorry'erdos_319.variants.isLittleO (N : ℕ) (c : ℕ → ℝ) (h : ∀ N, IsGreatest { (#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) } (c N)) : c =o[atTop] (answer(sorry) : ℕ → ℝ) := N:ℕc:ℕ → ℝh:∀ (N : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑(#A) = x} (c N)⊢ c =o[atTop] sorry All goals completed! 🐙

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.

@[category research solved, AMS 5] theorem declaration uses 'sorry'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) } := ⊢ ∃ o, o =o[atTop] 1 ∧ ∀ᶠ (N : ℕ) in atTop, (1 - 1 / rexp 1 + o N) * ↑N ≤ sSup {x | ∃ A, ∃ (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑(#A) = x} All goals completed! 🐙 end Erdos319