---

Claim. There are finite additive $2$-bases of $k$ positive elements whose range, the largest $n$ with $\{0,\ldots,n\}\subseteq A+A$, is at least $\frac87(\frac k2)^2+O(k)$ (equation (3), printed p. 118, derived from the basis $B_2$ of p. 121 with the parameters of p. 123). In the notation of Problem 791, with $n(k)$ the largest range of a $2$-basis of $k$ elements and $g(n)=\min\{k:n(k)\ge n\}$, the construction gives $n(k)\ge(\frac27-\varepsilon)k^2$ for all large $k$ (counting the zero element changes $k$ by one), so for large $n$ the integer $k=\lceil\sqrt{n/(2/7-\varepsilon)}\rceil$ has $n(k)\ge n$, hence $g(n)\le k$ and $g(n)^2\le(\frac72+o(1))n$, so

$$\limsup_{n\to\infty}\frac{g(n)}{\sqrt n}\le\sqrt{\tfrac72}<2$$

and the question whether $g(n)\sim2n^{1/2}$ has the answer no. Rohrbach's conjecture $n_2(k)=k^2/4+O(k)$, of which the site's question is the asymptotic form, is refuted with it. A. Mrose, Untere Schranken für die Reichweiten von Extremalbasen fester Ordnung, Abh. Math. Sem. Univ. Hamburg 48 (1979), no. 1, 118--124, cited as [Mr79] on the problem page. Library home mrose_1979_untere_schranken_reichweiten_extremalbasen_fester_ordnung; result page Equation (3). Mrose's $k$ counts the positive elements, so his $\frac87(\frac k2)^2$ is the $\liminf n(k)/k^2\ge2/7$ that Kohonen quotes for him. Hämmerer and Hofmeister refuted the guess independently and earlier in print, with the constant $5/18$ (their claim page); Mrose's paper does not cite them. Kohonen's basis raises the constant to $85/294$ (his claim page), the best upper bound for the estimate.

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 between $(2.181\ldots+o(1))n\le g(n)^2\le(3.458\ldots+o(1))n$.

Depends on.

• Mrose 1979, equation (3)

Acceptance. Refereed: the paper is the publisher's version of record in the Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, received 25 April 1975 and published in April 1979 (Crossref: published in print 1979-04, online 1979-04-01), which dates this page. The site's curator, Thomas F. Bloom, credits the disproof of $g(n)\sim2n^{1/2}$ to Mrose in the problem page's commentary (label OPEN, page last edited 24 September 2025); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. Kohonen's paper (p. 1) restates Mrose's bound as the previous record, a citation and not a review. Equation (3), the basis $B_2$ and Satz 2 are checked at statement depth; the parameter optimization behind (3) is not printed, and the proof is not reviewed in this corpus.