number-theory / Number Theory, Additive Bases
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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Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open
Research memory
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.
Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open
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