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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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Does there exist $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ which is a minimal basis of order $2$ (every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property) such that $\lim_{k\to \infty}a_k/k^2=c$ for some $c\neq 0$? A claimed construction gives a minimal basis with $A(x)=C\sqrt{x}+O(1)$, answering the question affirmatively; Erdős and Graham had conjectured a negative answer.

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