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import FormalConjecturesUtilGreen's Open Problem 27
References:
[Gr24] Green, Ben. "100 open problems." (2024).
[Be23] Bedert, Benjamin. "On unique sums in Abelian groups." Combinatorica 44.2 (2024): 269-298.
[St76] Straus, E. G. "Differences of residues (mod p)." Journal of Number Theory 8.1 (1976): 40-42.
open Asymptotics Filter
namespace Green27
This is $m(p)$ in [Be23]: the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation.
noncomputable def m (p : ℕ) : ℝ :=
(sInf { (A.card) | (A : Finset (ZMod p)) (_ : 2 ≤ A.card) (_ : HasNoUniqueRepresentation A) } : ℝ)
atTop restricted to prime numbers.
def primesAtTop : Filter ℕ := atTop ⊓ 𝓟 {p : ℕ | p.Prime}Best-known lower bound [Be23, Theorem 3].
noncomputable def lowerBest (p : ℕ) : ℝ :=
(Real.sqrt (Real.log (Real.log (Real.log (p : ℝ)))) /
Real.log (Real.log (Real.log (Real.log (p : ℝ))))) * Real.log (p : ℝ)Best-known upper bound [Be23, Theorem 5].
noncomputable def upperBest (p : ℕ) : ℝ := (Real.log (p : ℝ)) ^ 2
What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?
@[category research open, AMS 5 11]
theorem green_27.equivalent :
(answer(sorry) : ℕ → ℝ) ~[primesAtTop] m := ⊢ sorry ~[primesAtTop] m
All goals completed! 🐙Propose a better lower bound along primes.
@[category research open, AMS 5 11]
theorem green_27.lower :
let ans := (answer(sorry) : ℕ → ℝ)
(lowerBest =o[primesAtTop] ans) ∧ (ans =O[primesAtTop] m) := ⊢ let ans := sorry;
lowerBest =o[primesAtTop] ans ∧ ans =O[primesAtTop] m
All goals completed! 🐙Propose a better upper bound along primes.
@[category research open, AMS 5 11]
theorem green_27.upper :
let ans := (answer(sorry) : ℕ → ℝ)
(ans =o[primesAtTop] upperBest) ∧ (m =O[primesAtTop] ans) := ⊢ let ans := sorry;
ans =o[primesAtTop] upperBest ∧ m =O[primesAtTop] ans
All goals completed! 🐙
We have $m(p) \geq \omega(p) \log p$ for some function $\omega(p)$ tending to infinity [Be23, Theorem 3].
@[category research solved, AMS 5 11]
theorem green_27.variants.lower_be23 :
∃ ω : ℕ → ℝ, Tendsto ω primesAtTop atTop ∧
∀ᶠ p in primesAtTop,
ω p * Real.log (p : ℝ) ≤ m p := ⊢ ∃ ω, Tendsto ω primesAtTop atTop ∧ ∀ᶠ (p : ℕ) in primesAtTop, ω p * Real.log ↑p ≤ m p
All goals completed! 🐙Upper bound: $m(p) \ll (\log p)^2$ [Be23, Theorem 5].
@[category research solved, AMS 5 11]
theorem green_27.variants.upper_be23 :
m =O[primesAtTop] upperBest := ⊢ m =O[primesAtTop] upperBest
All goals completed! 🐙Previous best-known lower bound $\log p \ll m(p)$ from [St76].
@[category research solved, AMS 5 11]
theorem green_27.variants.previous_lower :
(fun p ↦ Real.log (p : ℝ)) =O[primesAtTop] m := ⊢ (fun p => Real.log ↑p) =O[primesAtTop] m
All goals completed! 🐙Previous best-known upper bound $m(p) \ll \sqrt{p}$ from [Be23].
@[category research solved, AMS 5 11]
theorem green_27.variants.previous_upper :
m =O[primesAtTop] (fun p ↦ Real.sqrt (p : ℝ)) := ⊢ m =O[primesAtTop] fun p => √↑p
All goals completed! 🐙
end Green27