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Claim. The Theorem of I. Z. Ruzsa, Sum-avoiding subsets, Ramanujan J. 9 (2005), no. 1--2, 77--82, display (1.1), printed p. 77: "$\frac2{\log3}\log n-1<l(n)\ll e^{c\sqrt{\log n}}$ with arbitrary $c>\sqrt{8\log2}$", where $\lambda(A)$ is the largest size of a subset $S\subset A$ with $s+s'\notin A$ for distinct $s,s'\in S$ and $l(n)=\min\{\lambda(A):A\subset\mathbb N,\ |A|=n\}$. A set of positive integers is a set of reals, so in the notation of Problem 787

$$g(n)\le l(n)\ll e^{c\sqrt{\log n}}\qquad(c>\sqrt{8\log2}),$$

with no reduction step. The upper estimate (§ 2, pp. 78--79) comes from a union of dilated lattice balls $U_r=\bigcup_{i<r}2^i(B_{r-i}+y)$ in $\mathbb Z^d$, in which more than $2^d$ points of one layer contain two whose sum lies in the next layer, projected to the positive integers in base $m$; Sanders describes the construction as Behrend's adapted. The lower half of the Theorem, $l(n)>\frac2{\log3}\log n-1$, is proved (pp. 79--81) by a greedy selection and improves the constant of Klarner's and Choi's logarithmic lower bounds. It bounds $l(n)$, over sets of positive integers, and reaches $g(n)$ only through Choi's reduction of the problem for real numbers to sets of integers (see Choi's page); it is superseded by Sanders's $(\log n)^{1+c}$. Cited as [Ru05] on the problem page. Library home ruzsa_2005_sum_avoiding_subsets; result page Theorem.

Covers. The upper bound $g(n)\ll e^{c\sqrt{\log n}}$ for every $c>\sqrt{8\log2}$, the site's $\exp(O(\sqrt{\log n}))$. Not covered: the order of growth of $g(n)$.

Depends on. No page of this wiki; the construction and the greedy argument are self-contained.

Acceptance. Refereed: the paper is the publisher's version of record in The Ramanujan Journal (received August 27, 2002, accepted December 23, 2002; Crossref record: issue of March 2005, with no day, so this page is named by the first of the month). The site's curator, Thomas F. Bloom, credits the upper bound to Ruzsa in the problem page's commentary (label OPEN, page last edited 23 January 2026); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The Theorem is checked against printed p. 77, the proof of the upper estimate was followed in full, and the proof of the lower estimate was read for its structure; none of this is an independent review.