FormalConjectures.ErdosProblems.«323»

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import FormalConjecturesUtil

Erdős Problem 323

Reference: erdosproblems.com/323


open Filter

open scoped Asymptotics


namespace Erdos323

Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers.

noncomputable def f (k m x : ℕ) : ℕ :=
  { n : ℕ | n ≤ x ∧ ∃ (v : Fin m → ℕ), n = ∑ i, v i ^ k }.ncard

Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$?

This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_323.parts.i :
    answer(sorry) ↔ ∀ k ≥ 1, ∀ ε > (0 : ℝ),
      (fun (x : ℕ) ↦ (x : ℝ) ^ (1 - ε)) =O[atTop] (fun (x : ℕ) ↦ (f k k x : ℝ)) := by⊢ True ↔ ∀ k ≥ 1, ∀ ε > 0, (fun x => ↑x ^ (1 - ε)) =O[atTop] fun x => ↑(f k k x)
  sorryAll goals completed! 🐙

Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_323.parts.ii :
    answer(sorry) ↔ ∀ k m : ℕ, 1 ≤ m → m < k →
      (fun (x : ℕ) ↦ (x : ℝ) ^ ((m : ℝ) / (k : ℝ))) =O[atTop] (fun (x : ℕ) ↦ (f k m x : ℝ)) := by⊢ True ↔ ∀ (k m : ℕ), 1 ≤ m → m < k → (fun x => ↑x ^ (↑m / ↑k)) =O[atTop] fun x => ↑(f k m x)
  sorryAll goals completed! 🐙

The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant $c>0$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_323.variants.k_eq_2 :
    ∃ c > 0, (fun (x : ℕ) ↦ (f 2 2 x : ℝ)) ~[atTop]
      (fun (x : ℕ) ↦ c * (x : ℝ) / Real.sqrt (Real.log (x : ℝ))) := by⊢ ∃ c > 0, (fun x => ↑(f 2 2 x)) ~[atTop] fun x => c * ↑x / √(Real.log ↑x)
  sorryAll goals completed! 🐙

For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_323.variants.k_gt_2 :
    answer(sorry) ↔ ∀ k > 2, (fun (x : ℕ) ↦ (f k k x : ℝ)) =o[atTop] (fun (x : ℕ) ↦ (x : ℝ)) := by⊢ True ↔ ∀ k > 2, (fun x => ↑(f k k x)) =o[atTop] fun x => ↑x
  sorryAll goals completed! 🐙


end Erdos323