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The claim. Theorem 4.5 of Y. O. Hamidoune and G. Zémor, On zero-free subset sums, Acta Arith. 78 (1996), no. 2, 143--152 (received 18 January 1996), p. 151: there is a function $\varepsilon(n)=O(n^{1/3}\ln n)$ such that for every subset $S$ of every finite abelian group $G$ of order $n$, $|S|>\sqrt{2n}+\varepsilon(n)$ implies that $0$ is the sum of a nonempty subset of $S$. Paged as Theorem 4.5 of Hamidoune and Zémor (1996); its prime form is Theorem 3.3 (p. 148), $|S|\ge\sqrt{2p}+5\ln p$. Applied to $G=\mathbb Z/N\mathbb Z$, the theorem gives the statement of Problem 540 for every large $N$ with any constant above $\sqrt2$, and the small $N$ are handled as on the problem page: below a fixed $n_0$, a constant $c\ge\sqrt{n_0}$ makes $c\sqrt N\ge N$, so only $A=\mathbb Z/N\mathbb Z$, which contains $0$, qualifies. The proof uses Olson's 1975 theorem that $|S|\ge3\sqrt{|G|}$ forces a zero sum in an abelian group $G$ (their Theorem 2.5, from J. E. Olson, Sums of sets of group elements, Acta Arith. 28 (1975), 147--156).

Acceptance. Refereed: the journal publication. Reviewed: the site's curator, Thomas Bloom, who is independent of the authors, labels the problem PROVED (LEAN) and his commentary credits this paper with the threshold $(1+o(1))\sqrt{2N}$ for abelian groups of order $N$.

Depends on. No page of this wiki; the proof's input from Olson (1975) is a published theorem cited above.