FormalConjectures.ErdosProblems.«52»

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import FormalConjecturesUtil

Erdős Problem 52

Reference: erdosproblems.com/52

open scoped Pointwisenamespace Erdos52

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$

@[category research open, AMS 11]
theorem erdos_52 : answer(sorry) ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ (C : ℝ), 0 < C ∧ ∀ (A : Finset ℤ),
    (max (A + A).card (A * A).card : ℝ) ≥ C * (A.card : ℝ) ^ (2 - ε) := by⊢ True ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ C, 0 < C ∧ ∀ (A : Finset ℤ), max ↑(A + A).card ↑(A * A).card ≥ C * ↑A.card ^ (2 - ε)
  sorryAll goals completed! 🐙

-- TODO(firsching): Add additional material.

end Erdos52