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Merged
YaelDillies merged 3 commits into from
Jan 18, 2026
Merged

fix(ErdosProblems/33) #1573

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0f4384e
Update 33.lean
CoolRmal Jan 8, 2026
8c4b738
Update 33.lean
CoolRmal Jan 8, 2026
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Update 33.lean
CoolRmal Jan 8, 2026
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14 changes: 6 additions & 8 deletions FormalConjectures/ErdosProblems/33.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -38,13 +38,11 @@ def AdditiveBasisCondition (A : Set ℕ) : Prop :=

/-- Let `A ⊆ ℕ` be a set such that every integer can be written as `n^2 + a`
for some `a` in `A` and `n ≥ 0`. What is the smallest possible value of
`lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2) = 0`?
`lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)`?
-/
@[category research open, AMS 11]
theorem erdos_33 : IsLeast
{ c : EReal | ∃ (A : Set ℕ), AdditiveBasisCondition A ∧
Filter.atTop.limsup (fun N => (A.interIcc 1 N).ncard / √N) = c}
answer(sorry) := by
theorem erdos_33 : ⨅ A : {A : Set ℕ | AdditiveBasisCondition A}, Filter.atTop.limsup (fun N =>
(A.1.interIcc 1 N).ncard / (√N : EReal)) = answer(sorry) := by
sorry

/--
Expand All @@ -56,14 +54,14 @@ theorem erdos_33.variants.one_mem_lowerBounds : ∃ A, AdditiveBasisCondition A
sorry

/--
The smallest possible value of `lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2) = 0`
The smallest possible value of `lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)`
is at most `2φ^(5/2) ≈ 6.66`, with `φ` equal to the golden ratio. Proven by
Wouter van Doorn.
-/
@[category research solved, AMS 11]
theorem erdos_33.variants.vanDoorn :
↑(2 * (φ ^ ((5 : ℝ) / 2))) ∈ lowerBounds { c : EReal | ∃ (A : Set ℕ), AdditiveBasisCondition A ∧
Filter.atTop.limsup (fun N => (A.interIcc 1 N).ncard / (√N)) = c} := by
⨅ A : {A : Set ℕ | AdditiveBasisCondition A}, Filter.atTop.limsup (fun N =>
(A.1.interIcc 1 N).ncard / (√N : EReal)) ≤ ↑(2 * (φ ^ ((5 : ℝ) / 2))) := by
sorry

end Erdos33
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