FormalConjectures.GreensOpenProblems.«33»

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import FormalConjecturesUtil

Ben Green's Open Problem 33

References:

[Gr24] Ben Green's Open Problem 33

[CaHa20] Caprace, Pierre-Emmanuel, and Pierre de la Harpe. "Groups with irreducibly unfaithful subsets for unitary representations." Confluentes Mathematici 12.1 (2020): 31-68.

[CrLe07] Croot, Ernie, and Vsevolod F. Lev. "Open problems in additive combinatorics." Additive combinatorics 43.207-233 (2007): 1.


open Filter

open scoped Pointwise


namespace Green33

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'green_33 :
    answer(sorry) ↔
      ∀ ε : ℝ, 0 < ε →
        ∃ᶠ q : ℕ+ in atTop,
          ∃ A : Finset (ZMod q),
            A + A = Finset.univ ∧
            |((A.card : ℝ) / Real.sqrt q - Real.sqrt 2)| < ε := by⊢ True ↔ ∀ (ε : ℝ), 0 < ε → ∃ᶠ (q : ℕ+) in atTop, ∃ A, A + A = Finset.univ ∧ |↑A.card / √↑↑q - √2| < ε
  sorryAll goals completed! 🐙

Trivial lower bound: if $A + A = \mathbb{Z}/q\mathbb{Z}$, then $|A|^2 \geq q$, since the sumset $A + A$ has at most $|A|^2$ elements.

@[category test, AMS 5 11]
theorem green_33.sanity_sq_bound (q : ℕ+) (A : Finset (ZMod q))
    (hA : A + A = Finset.univ) : (q : ℕ) ≤ A.card ^ 2 := byq:ℕ+A:Finset (ZMod ↑q)hA:A + A = Finset.univ⊢ ↑q ≤ A.card ^ 2
  calc (q : ℕ) = Fintype.card (ZMod q) := (ZMod.card q).symm
    _ = (Finset.univ : Finset (ZMod q)).card := Finset.card_univ.symm
    _ = (A + A).card := byq:ℕ+A:Finset (ZMod ↑q)hA:A + A = Finset.univ⊢ Finset.univ.card = (A + A).card rw [hA]All goals completed! 🐙
    _ ≤ A.card * A.card := Finset.card_add_le ..
    _ = A.card ^ 2 := (sq A.card).symm


-- TODO(jgd): Add variants from comments in [Gr24]

end Green33