FormalConjectures.ErdosProblems.«156»

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import FormalConjecturesUtil

Erdős Problem 156

References:

erdosproblems.com/156

[ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347.

[Ru98b] Ruzsa, Imre Z., A small maximal Sidon set. Ramanujan J. (1998), 55-58.

open Finset Filternamespace Erdos156

The size of the smallest maximal Sidon set in ${1, \dots, N}$.

noncomputable def minMaximalSidonSet (N : ℕ) : ℕ :=
  open scoped Classical in
  sInf (((Icc 1 N).powerset.filter fun (A : Finset ℕ) ↦
    Set.IsMaximalSidonSetIn (A : Set ℕ) N).image card : Set ℕ)

Does there exist a maximal Sidon set $A\subset {1,\ldots,N}$ of size $O(N^{1/3})$?

A question of Erdős, Sárközy, and Sós [ESS94].

@[category research open, AMS 5]
theorem erdos_156 :
    answer(sorry) ↔
      (fun N ↦ (minMaximalSidonSet N : ℝ)) =O[atTop] (fun N ↦ (N : ℝ) ^ (1 / 3 : ℝ)) := by⊢ True ↔ (fun N ↦ ↑(minMaximalSidonSet N)) =O[atTop] fun N ↦ ↑N ^ (1 / 3)
  sorryAll goals completed! 🐙

@[category test, AMS 5]
theorem greedySidonSet_isSidon (n : ℕ) : IsSidon (Finset.greedySidonBelow n : Set ℕ) := byn:ℕ⊢ IsSidon ↑(greedySidonBelow n)
  intro i₁ hi₁ j₁ hj₁ i₂ hi₂ j₂ hj₂ eqn:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕhi₂:i₂ ∈ ↑(greedySidonBelow n)j₂:ℕhj₂:j₂ ∈ ↑(greedySidonBelow n)eq:i₁ + i₂ = j₁ + j₂⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁
  have subset : Finset.greedySidonBelow n ⊆ (Finset.greedySidon.aux n).1 := Finset.filter_subset _ _n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕhi₂:i₂ ∈ ↑(greedySidonBelow n)j₂:ℕhj₂:j₂ ∈ ↑(greedySidonBelow n)eq:i₁ + i₂ = j₁ + j₂subset:greedySidonBelow n ⊆ ↑(greedySidon.aux n).1⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁
  exact (Finset.greedySidon.aux n).1.2 i₁ (subset hi₁) j₁ (subset hj₁) i₂ (subset hi₂) j₂ (subset hj₂) eqAll goals completed! 🐙

It is easy to prove that the greedy construction of a maximal Sidon set in ${1,\ldots,N}$ has size $\gg N^{1/3}$.

@[category research solved, AMS 5]
theorem erdos_156.variants.greedy_lower_bound :
    (fun N ↦ ((Finset.greedySidonBelow N).card : ℝ)) ≫ (fun N ↦ (N : ℝ) ^ (1 / 3 : ℝ)) := by⊢ (fun N ↦ ↑N ^ (1 / 3)) =O[atTop] fun N ↦ ↑(#(greedySidonBelow N))
  sorryAll goals completed! 🐙

Ruzsa [Ru98b] constructed a maximal Sidon set of size $\ll (N\log N)^{1/3}$.

@[category research solved, AMS 5]
theorem erdos_156.variants.ruzsa_upper_bound :
    (fun N ↦ (minMaximalSidonSet N : ℝ)) ≪
      (fun N ↦ ((N : ℝ) * Real.log N) ^ (1 / 3 : ℝ)) := by⊢ (fun N ↦ ↑(minMaximalSidonSet N)) =O[atTop] fun N ↦ (↑N * Real.log ↑N) ^ (1 / 3)
  sorryAll goals completed! 🐙

end Erdos156