FormalConjectures.ErdosProblems.«14»

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import FormalConjecturesUtil

Erdős Problem 14

Reference: erdosproblems.com/14

namespace Erdos14open Asymptotics Filter

The number of integers in ${1,\ldots,N}$ which are not representable in exactly one way as the sum of two elements from $A$ (either because they are not representable at all, or because they are representable in more than one way).

noncomputable def nonUniqueSumCount (A : Set ℕ) (N : ℕ) : ℝ :=
  ((Set.Icc 1 N) \ (allUniqueSums A)).ncard

noncomputable def almostSquareRoot (ε : ℝ) (N : ℕ) : ℝ :=
  N ^ (1/2 - ε)

noncomputable def squareRoot (N : ℕ) : ℝ :=
  Real.sqrt N

Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon > 0$ and large $N$, $|{1,\ldots,N} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?

@[category research open, AMS 11]
theorem erdos_14.parts.i :
    answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε := by⊢ True ↔ ∀ (A : Set ℕ), ∀ ε > 0, almostSquareRoot ε =O[atTop] nonUniqueSumCount A sorryAll goals completed! 🐙

Is it possible that $|{1,\ldots,N} \setminus B| = o(N^\frac{1}{2})$?

@[category research open, AMS 11]
theorem erdos_14.parts.ii :
    answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot := by⊢ True ↔ ∃ A, nonUniqueSumCount A =o[atTop] squareRoot
  sorryAll goals completed! 🐙

end Erdos14