FormalConjectures.ErdosProblems.«512»

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import FormalConjecturesUtil

Erdős Problem 512

References:

erdosproblems.com/512

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Ko81] Konyagin, S. V., On the Littlewood problem. Izv. Akad. Nauk SSSR Ser. Mat. (1981), 243-265, 463.

[MPS81] McGehee, O. Carruth and Pigno, Louis and Smith, Brent, Hardy's inequality and the $L^1$ norm of exponential sums. Ann. of Math. (2) (1981), 613-618.

open scoped ExponentialSumnamespace Erdos512

Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then $$\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$$ where $e(x)=e^{2\pi ix }$?

Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81].

@[category research solved, AMS 11 42, formal_proof using lean4 at "https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/512/Erdos512.lean"]
theorem erdos_512 : answer(True) ↔
    ∃ c > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℤ), A.card = N →
      c * Real.log N ≤ ∫ θ in (0 : ℝ)..1, ‖∑ n ∈ A, e (n * θ)‖ := by⊢ True ↔ ∃ c > 0, ∀ (N : ℕ) (A : Finset ℤ), A.card = N → c * Real.log ↑N ≤ ∫ (θ : ℝ) in 0..1, ‖∑ n ∈ A, e (↑n * θ)‖
  sorryAll goals completed! 🐙

end Erdos512