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declared open · formalizederdos:326
Does there exist A={a1<a2<⋯ }⊂NA=\{a_1<a_2<\cdots\}\subset \mathbb{N} which is a minimal basis of order 22 (every large integer is the sum of 22 elements from AA, and no proper subset of AA has this property) such that lim⁡k→∞ak/k2=c\lim_{k\to \infty}a_k/k^2=c for some c≠0c\neq 0? A claimed construction gives a minimal basis with A(x)=Cx+O(1)A(x)=C\sqrt{x}+O(1), answering the question affirmatively; Erdős and Graham had conjectured a negative answer.
number theoryadditive basisN/A4 sources
declared proved (Lean) · formalizederdos:330
True ↔ ∃ A h, Erdos330.MinAsymptoticAddBasisOfOrder A h ∧ 0 < A.upperDensity ∧ ∀ n ∈ A, 0 < (Erdos330.UnrepWithout A n h).upperDensity
number theoryadditive basisN/A5 sources
declared disproved (Lean) · formalizederdos:333
False ↔ ∀ (A : Set ℕ), A.HasDensity 0 → ∃ B, A ⊆ B + B ∧ (fun N => ↑(B ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => √↑N
number theoryadditive basisN/A6 sources
declared openerdos:336
If h(r)h(r) is the maximal finite exact order attainable by an additive basis of order at most rr, what is lim⁡r→∞h(r)/r2\lim_{r \to \infty} h(r)/r^2? The candidate proof identifies the sharp limit 1/31/3.
number theoryadditive basispossible2 sources
declared openerdos:870
Let k≥3k\geq 3 and AA be an additive basis of order kk. Does there exist a constant c=c(k)>0c=c(k)>0 such that if r(n)≥clog⁡nr(n)\geq c\log n for all large nn (where r(n)r(n) counts representations of nn as a sum of at most kk elements of AA) then AA must contain a minimal basis of order kk? The claimed answer is no, for every k≥3k\geq 3.
number theoryadditive basisN/A3 sources
declared disproved (Lean) · formalizederdos:871
False ↔ ∀ (A : Set ℕ), ((∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ A, ∃ b ∈ A, a + b = n) ∧ ∀ (t : ℕ), ∀ᶠ (n : ℕ) in Filter.atTop, ∃ pairs, pairs.card ≥ t ∧ ∀ p ∈ pairs, p.1 ∈ A ∧ p.2 ∈ A ∧ p.1 + p.2 = n ∧ p.1 ≤ p.2) → ∃ B C, (∀ (x : ℕ), x ∈ A ↔ x ∈ B ∨ x ∈ C) ∧ Disjoint B C ∧ (∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ B, ∃ b ∈ B, a + b = n) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ C, ∃ b ∈ C, a + b = n
number theoryadditive basisN/A6 sources

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