FormalConjectures.ErdosProblems.«39»

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import FormalConjecturesUtil

Erdős Problem 39

Reference: erdosproblems.com/39


namespace Erdos39


open Filter

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap {1\ldots,N}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_39 : answer(sorry) ↔ ∃ (A : Set ℕ), A.Infinite ∧ IsSidon A ∧
    ∀ᵉ  (ε  > (0 : ℝ)),
    (· ^ (1 / 2 - ε) : ℕ → ℝ) =O[atTop] fun N => (((Set.Icc 1 N) ∩ A).ncard : ℝ) := by⊢ True ↔ ∃ A, A.Infinite ∧ IsSidon A ∧ ∀ ε > 0, (fun x => ↑x ^ (1 / 2 - ε)) =O[atTop] fun N => ↑(Set.Icc 1 N ∩ A).ncard
  sorryAll goals completed! 🐙


-- TODO(firsching): add the various known bounds as variants.

end Erdos39