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Claim. Kelly proves two theorems on the restricted order, in the sense of Problem 338, of bases of order $2$. First, let $A\subseteq\mathbb{N}\cup\{0\}$ be a basis of order $2$ in the classical sense: every nonnegative integer is a sum of two elements of $A$. Then $A$ has a restricted order, at most $4$: every large integer is a sum of at most four distinct elements of $A$. Second, let $A$ be an asymptotic basis of order $2$, every large integer being a sum of at most two elements of $A$, whose counting function satisfies $A(x)\ge Cx/\log\log x$ for some $C>0$ and all large $x$. Then $A$ has a restricted order, at most $3$. Positive lower density, the hypothesis in the site's remarks, is a special case of this one. Kelly conjectures that $3$ holds for every basis of order $2$. The paper is Kelly, John B., Restricted bases, Amer. J. Math. 79 (1957), no. 2, 258--264, whose theorems are stated here as the zbMATH review Zbl 0077.26304 gives them. The site's remarks state the bound $4$ for asymptotic bases of order $2$ and credit it to Kelly; that statement is Hennecart's theorem of 2005, recorded at Hennecart 2005, which also refutes Kelly's conjecture with a basis of restricted order exactly $4$, so Kelly's bound $4$ is attained.

Covers. Classical bases of order $2$, and asymptotic bases of order $2$ whose counting function is at least $Cx/\log\log x$. For both classes the statement's first question is settled, since a restricted order always exists and no further condition is needed, and its second question is settled with the bound $4$, which Hennecart's example shows is best possible for classical bases, and the bound $3$ for the second class. The claim's value is answered because the result determines what the first question asks to determine for these classes and proves a bound for the second. Nothing is covered for other asymptotic bases of order $2$, which Hennecart's theorem treats, nor for bases of order $3$ or more, where Bateman's example in the site's remarks, $\{1\}\cup\{x>0:h\mid x\}$, has order $h\ge3$ and no restricted order, nor for the statement's third question, the conditions under which the restricted order equals the order.

Acceptance. The refereed evidence is the journal publication cited above, in the American Journal of Mathematics. The site's curator cites the paper in the remarks of a problem the site labels OPEN, which credits the partial result without settling the problem, so no reviewed evidence is listed. The record gives the issue month, April 1957, and no finer date, so the page is dated to the first day of that month.

Depends on. Nothing in this wiki.