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Claim. J. Kohonen, An improved lower bound for finite additive 2-bases, J. Number Theory 174 (2017), 518--524 (arXiv:1606.04770), equation (1), p. 1. A set $A$ of non-negative integers is an additive $2$-basis of size $k=|A|$ (the zero counted) and range $n(A)$ if $A+A$ contains $0,1,\ldots,n(A)$ but not $n(A)+1$; with $n(k)$ the maximal range over bases of size $k$, the paper proves

$$\liminf_{k\to\infty}\frac{n(k)}{k^2}\ge\frac{85}{294}>0.2891,$$

by a generalized Mrose basis built from three elementary segments placed at multiples of $t^2$ (Theorem 1, pp. 3--4). In the notation of Problem 791, $g(n)=\min\{k:n(k)\ge n\}$; if $n(k)\ge(85/294-\varepsilon)k^2$ for all $k\ge k_0(\varepsilon)$, then for large $n$ the integer $k=\lceil\sqrt{n/(85/294-\varepsilon)}\rceil$ has $n(k)\ge n$, so $g(n)\le k$ and

$$g(n)^2\le\Bigl(\frac{294}{85}+o(1)\Bigr)n,\qquad\frac{294}{85}=3.4588\ldots,$$

the upper bound the site's commentary gives as $3.458\cdots$. Since $294/85<4$, the bound also gives $\limsup g(n)/\sqrt n<2$, so the guess $g(n)\sim2n^{1/2}$ is false; that refutation was first in print in Hämmerer and Hofmeister's paper and is also Mrose's.

Covers. The upper bound $g(n)^2\le(294/85+o(1))n$ for the estimate of $g(n)$. Not covered: the value of $\lim g(n)^2/n$, or whether it exists.

Depends on.

• Kohonen 2017, equation (1)

Acceptance. Refereed: the paper is published in the Journal of Number Theory (Crossref: 2017-05); the preprint was first posted on 2016-06-15, which dates this page. The site's curator, Thomas F. Bloom, cites the bound in the problem's commentary, but the site labels the problem OPEN, so the citation is not listed as reviewed. The placement of Theorem 1 that gives $85/294$ is not checked in this corpus.