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import FormalConjecturesUtilErdős Problem 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 Filteropen scoped Classical
namespace Erdos156
The size of the smallest maximal Sidon set in ${1, \dots, N}$.
noncomputable def minMaximalSidonSet (N : ℕ) : ℕ :=
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)) := ⊢ True ↔ (fun N => ↑(minMaximalSidonSet N)) =O[atTop] fun N => ↑N ^ (1 / 3)
All goals completed! 🐙
@[category test, AMS 5]
theorem greedySidonSet_isSidon (n : ℕ) : IsSidon (Finset.greedySidonBelow n : Set ℕ) := n:ℕ⊢ IsSidon ↑(greedySidonBelow n)
intro i₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)⊢ ∀ j₁ ∈ ↑(greedySidonBelow n),
∀ i₂ ∈ ↑(greedySidonBelow n), ∀ j₂ ∈ ↑(greedySidonBelow n), i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕ⊢ j₁ ∈ ↑(greedySidonBelow n) →
∀ i₂ ∈ ↑(greedySidonBelow n), ∀ j₂ ∈ ↑(greedySidonBelow n), i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)⊢ ∀ i₂ ∈ ↑(greedySidonBelow n), ∀ j₂ ∈ ↑(greedySidonBelow n), i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕ⊢ i₂ ∈ ↑(greedySidonBelow n) → ∀ j₂ ∈ ↑(greedySidonBelow n), i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕhi₂:i₂ ∈ ↑(greedySidonBelow n)⊢ ∀ j₂ ∈ ↑(greedySidonBelow n), i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕhi₂:i₂ ∈ ↑(greedySidonBelow n)j₂:ℕ⊢ j₂ ∈ ↑(greedySidonBelow n) → i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ n:ℕi₁:ℕhi₁:i₁ ∈ ↑(greedySidonBelow n)j₁:ℕhj₁:j₁ ∈ ↑(greedySidonBelow n)i₂:ℕhi₂:i₂ ∈ ↑(greedySidonBelow n)j₂:ℕhj₂:j₂ ∈ ↑(greedySidonBelow n)⊢ i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ 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 + j₂⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁
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 := filter_subset (fun x => x ≤ n) ↑(greedySidon.aux n).1⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁
All 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)) := ⊢ (fun N => ↑N ^ (1 / 3)) =O[atTop] fun N => ↑(#(greedySidonBelow N))
All 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)) := ⊢ (fun N => ↑(minMaximalSidonSet N)) =O[atTop] fun N => (↑N * Real.log ↑N) ^ (1 / 3)
All goals completed! 🐙
end Erdos156