FormalConjectures.ErdosProblems.«1109»

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import FormalConjecturesUtil

Erdős Problem 1109

References:

erdosproblems.com/1109

[ErSa87] P. Erdős and A. Sárközy, On divisibility properties of integers of the form $a+a'$, Acta Math. Hungar. (1987), 117--122.

[Gy01] Katalin Gyarmati, On divisibility properties of integers of the form $ab+1$, Period. Math. Hungar. (2001), 71--79.

[Ko04] S. V. Konyagin, Problems of the set of square-free numbers, Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.

[Sa92c] G. N. Sárközy, On a problem of P. Erdős, Acta Math. Hungar. (1992), 271--282.

open Filter Asymptoticsopen scoped Pointwisenamespace Erdos1109

f N is the largest size of a subset A ⊆ {1, ..., N} such that every element of A + A is squarefree.

noncomputable def f (N : ℕ) : ℕ :=
  sSup {k : ℕ | ∃ A : Finset ℕ,
    A ⊆ Finset.Icc 1 N ∧ (∀ n ∈ A + A, Squarefree n) ∧ A.card = k}

Let $f(N)$ be the size of the largest subset $A\subseteq {1,\ldots,N}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\leq N^{o(1)}$, or even $f(N) \leq (\log N)^{O(1)}$?

This theorem formalizes the subpolynomial bound as f(N) = O(N^ε) for every ε > 0.

@[category research open, AMS 5 11]
theorem erdos_1109 :
    answer(sorry) ↔ ∀ ε > (0 : ℝ),
      (fun N : ℕ => (f N : ℝ)) ≪ fun N : ℕ => (N : ℝ) ^ ε := by⊢ True ↔ ∀ ε > 0, (fun N ↦ ↑(f N)) =O[atTop] fun N ↦ ↑N ^ ε
  sorryAll goals completed! 🐙

Is the stronger polylogarithmic bound $f(N) \leq (\log N)^{O(1)}$ true?

@[category research open, AMS 5 11]
theorem erdos_1109.variants.polylog :
    answer(sorry) ↔ ∃ C > (0 : ℝ),
      (fun N : ℕ => (f N : ℝ)) ≪ fun N : ℕ => (Real.log N) ^ C := by⊢ True ↔ ∃ C > 0, (fun N ↦ ↑(f N)) =O[atTop] fun N ↦ Real.log ↑N ^ C
  sorryAll goals completed! 🐙

Erdős and Sárközy [ErSa87] proved the lower bound $\log N \ll f(N)$.

@[category research solved, AMS 5 11]
theorem erdos_1109.variants.erdos_sarkozy_lower :
    (fun N : ℕ => Real.log N) ≪ fun N : ℕ => (f N : ℝ) := by⊢ (fun N ↦ Real.log ↑N) =O[atTop] fun N ↦ ↑(f N)
  sorryAll goals completed! 🐙

Erdős and Sárközy [ErSa87] proved the upper bound $f(N) \ll N^{3/4}\log N$.

@[category research solved, AMS 5 11]
theorem erdos_1109.variants.erdos_sarkozy_upper :
    (fun N : ℕ => (f N : ℝ)) ≪
      fun N : ℕ => (N : ℝ) ^ ((3 : ℝ) / 4) * Real.log N := by⊢ (fun N ↦ ↑(f N)) =O[atTop] fun N ↦ ↑N ^ (3 / 4) * Real.log ↑N
  sorryAll goals completed! 🐙

Konyagin [Ko04] improved the lower bound to $\log\log N(\log N)^2 \ll f(N)$.

@[category research solved, AMS 5 11]
theorem erdos_1109.variants.konyagin_lower :
    (fun N : ℕ => Real.log (Real.log N) * (Real.log N) ^ 2) ≪
      fun N : ℕ => (f N : ℝ) := by⊢ (fun N ↦ Real.log (Real.log ↑N) * Real.log ↑N ^ 2) =O[atTop] fun N ↦ ↑(f N)
  sorryAll goals completed! 🐙

Konyagin [Ko04] improved the upper bound to $f(N) \ll N^{11/15+o(1)}$.

@[category research solved, AMS 5 11]
theorem erdos_1109.variants.konyagin_upper :
    ∀ ε > (0 : ℝ), (fun N : ℕ => (f N : ℝ)) ≪
      fun N : ℕ => (N : ℝ) ^ ((11 : ℝ) / 15 + ε) := by⊢ ∀ ε > 0, (fun N ↦ ↑(f N)) =O[atTop] fun N ↦ ↑N ^ (11 / 15 + ε)
  sorryAll goals completed! 🐙

end Erdos1109