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Claim. N. Hämmerer and G. Hofmeister, Zu einer Vermutung von Rohrbach, J. Reine Angew. Math. 286/287 (1976), 239--247, inequality (1), printed p. 241. A set $\mathfrak A$ of non-negative integers is an Abschnittsbasis of order $2$ for $n$ if every integer of $[0,n]$ is a sum of two elements of $\mathfrak A$; its range $n(2,\mathfrak A)$ is the largest such $n$, and $n(2,k)$ is the largest range over bases with $k$ positive elements (p. 239; the zero is not counted). The paper proves

$$n(2,k)>\frac{10}9\cdot\frac{k^2}4=\frac{5}{18}k^2\qquad\text{for all }k\ge1,$$

and concludes that $n(2,k)\sim\frac14k^2$ fails (p. 241), refuting Rohrbach's conjecture as the paper states it on p. 240. In the notation of Problem 791: a basis of $k$ positive elements also contains $0$, so one with range at least $n$, with its elements above $n$ dropped, is a set of at most $k+1$ elements of $\{0,\ldots,n\}$ whose pairwise sums cover $\{0,\ldots,n\}$; hence $g(n)\le k+1$ whenever $n\le n(2,k)$. For large $n$ the integer $k=\lceil\sqrt{18n/5}\,\rceil$ has $n(2,k)>n$, so $g(n)\le k+1$, $g(n)^2\le(\frac{18}5+o(1))n$ and

$$\limsup_{n\to\infty}\frac{g(n)}{\sqrt n}\le\sqrt{3.6}<2,$$

so the question whether $g(n)\sim2n^{1/2}$ has the answer no.

Covers. The "in particular" question only: $g(n)\sim2n^{1/2}$ is false. Not covered: the estimate of $g(n)$, which the problem page records as open.

Depends on. No page of this wiki; the construction is self-contained.

Acceptance. Refereed: the paper is published in the Journal für die reine und angewandte Mathematik (Crossref: issued 1976-11-01), which dates this page; zbMATH reviews it as Zbl 0332.10032. It appeared in print before Mrose's refutation (its claim page), which was received in 1975, published in 1979 and does not cite it; the two refutations are independent. The site does not cite this paper. The construction behind (1) is not reviewed in this corpus.