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Erdős problem 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.

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FormalConjectures/ErdosProblems/

326.lean

Retained formal statement1 of 2

Let A⊂NA \subset \mathbb{N} be an additive basis of order 2.

Must there exist B={b1<b2<… }⊆AB = \{b_1 < b_2 < \dots\} \subseteq A which is also a basis such that lim⁡k→∞bkk2\lim_{k\to\infty} \frac{b_k}{k^2} does not exist?

FormalConjectures/ErdosProblems/326.leanErdos326.erdos_3268 linesExact file
True ↔  ∀ (A : Set ℕ),    A.IsAddBasisOfOrder 2 →      ∃ b,        StrictMono b ∧          ∀ (n : ℕ),            b n ∈ A ∧              (Set.range b).IsAddBasis ∧ ∀ (x : ℝ), ¬Filter.Tendsto (fun n => ↑(b n) / ↑n ^ 2) Filter.atTop (nhds x)
OpenStatement only, no proof