FormalConjectures.ErdosProblems.«325»

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import FormalConjecturesUtil

Erdős Problem 325

Reference: erdosproblems.com/325


open Asymptotics Filter


namespace Erdos325

A predicate for $n$ to be the sum of three $k$th powers.

def IsSumThreePower (k n : ℕ) : Prop := ∃ a b c, a ^ k + b ^ k + c ^ k = n

The number of integers $\leq x$ which are the sum of three $k$th powers.

noncomputable def cardIsSumThreePowerBelow (k x : ℕ) : ℕ :=
  {n ∈ Set.Iic x | IsSumThreePower k n}.ncard

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_325 :
     answer(sorry) ↔ ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ (3 / k : ℝ)) =O[atTop]
      (fun x : ℕ => (cardIsSumThreePowerBelow k x : ℝ)) := by⊢ True ↔ ∀ (k : ℕ), 3 ≤ k → (fun x => ↑x ^ (3 / ↑k)) =O[atTop] fun x => ↑(cardIsSumThreePowerBelow k x)
  sorryAll goals completed! 🐙

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_325.variants.weaker :
    answer(sorry) ↔ ∀ ε > 0, ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ ((3 / k : ℝ) - ε)) =O[atTop]
      (fun x => (cardIsSumThreePowerBelow k x : ℝ)) := by⊢ True ↔ ∀ ε > 0, ∀ (k : ℕ), 3 ≤ k → (fun x => ↑x ^ (3 / ↑k - ε)) =O[atTop] fun x => ↑(cardIsSumThreePowerBelow k x)
  sorryAll goals completed! 🐙

For $k = 3$, the best known is due to Wooley [Wo15] [Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_325.variants.wooley :
    (fun x : ℕ => (x : ℝ) ^ (0.917 : ℝ)) =O[atTop] (fun x => (cardIsSumThreePowerBelow 3 x : ℝ)) := by⊢ (fun x => ↑x ^ 0.917) =O[atTop] fun x => ↑(cardIsSumThreePowerBelow 3 x)
  sorryAll goals completed! 🐙


end Erdos325