FormalConjectures.ErdosProblems.«153»

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import FormalConjecturesUtil

Erdős Problem 153

#TODO: Formalize the corresponding conjecture for infinite Sidon sets.

References:

erdosproblems.com/153

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


open scoped Pointwise

open Filter Finset Nat


namespace Erdos153

Define $f(n)$ to be the minimum of $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2$ as $A$ ranges over all Sidon sets of size $n$, where $A+A={s_1<\cdots<s_t}$.

noncomputable def f (n : ℕ) : ℝ := ⨅ A : {A : Finset ℕ | A.card = n ∧ IsSidon (A : Set ℕ)},
  let s := (A.1 + A).orderIsoOfFin rfl
  (∑ i : Set.Ico 1 ((A.1 + A).card), (s ⟨i, i.2.2⟩ - s ⟨i - 1, byn:ℕA:↑{A | #A = n ∧ IsSidon ↑A}s:Fin #(↑A + ↑A) ≃o ↥(↑A + ↑A) := (↑A + ↑A).orderIsoOfFin ⋯i:↑(Set.Ico 1 #(↑A + ↑A))⊢ ↑i - 1 < #(↑A + ↑A) grindAll goals completed! 🐙⟩) ^ 2 : ℝ) / ((A.1 + A).card : ℝ)

Let $A$ be a finite Sidon set and $A+A={s_1<\cdots<s_t}$. Is it true that $$\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$$ as $\lvert A\rvert\to \infty$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_153 : answer(sorry) ↔ Tendsto f atTop atTop := by⊢ True ↔ Tendsto f atTop atTop
  sorryAll goals completed! 🐙


end Erdos153