FormalConjectures.ErdosProblems.«818»

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import FormalConjecturesUtil

Erdős Problem 818

References:

erdosproblems.com/818

[So09d] Solymosi, József, Bounding multiplicative energy by the sumset. Adv. Math. (2009), 402-408.

open scoped Pointwisenamespace Erdos818

Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}$$ for some constant $C>0$?

This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$ See also [52].

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos818.lean"]
theorem erdos_818 : answer(True) ↔
    ∀ K : ℝ, 0 < K → ∃ C : ℝ, 0 < C ∧ ∃ c : ℝ, 0 < c ∧
      ∀ A : Finset ℤ, 2 ≤ A.card → ((A + A).card : ℝ) ≤ K * (A.card : ℝ) →
        c * (A.card : ℝ) ^ 2 / (Real.log (A.card : ℝ)) ^ C ≤ ((A * A).card : ℝ) := by⊢ True ↔
  ∀ (K : ℝ),
    0 < K →
      ∃ C,
        0 < C ∧
          ∃ c,
            0 < c ∧
              ∀ (A : Finset ℤ),
                2 ≤ A.card → ↑(A + A).card ≤ K * ↑A.card → c * ↑A.card ^ 2 / Real.log ↑A.card ^ C ≤ ↑(A * A).card
  sorryAll goals completed! 🐙

This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$

@[category research solved, AMS 5 11]
theorem erdos_818.variants.solymosi :
    ∀ K : ℝ, 0 < K → ∃ c : ℝ, 0 < c ∧
      ∀ A : Finset ℤ, 2 ≤ A.card → ((A + A).card : ℝ) ≤ K * (A.card : ℝ) →
        c * (A.card : ℝ) ^ 2 / Real.log (A.card : ℝ) ≤ ((A * A).card : ℝ) := by⊢ ∀ (K : ℝ),
  0 < K →
    ∃ c,
      0 < c ∧
        ∀ (A : Finset ℤ), 2 ≤ A.card → ↑(A + A).card ≤ K * ↑A.card → c * ↑A.card ^ 2 / Real.log ↑A.card ≤ ↑(A * A).card
  sorryAll goals completed! 🐙

end Erdos818