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Statement

For a finite abelian group $G$ and $S\subset G$, $\Sigma^*(S)$ is the set of sums of nonempty subsets of $S$ (p. 143). Theorem 3.3 (p. 148). Let $G$ be a group of prime order $p$ and let $S\subset G$. Then

$$|S|\ge\sqrt{2p}+5\ln p\quad\text{implies}\quad0\in\Sigma^*(S).$$

Source. Y. O. Hamidoune and G. Zémor, On zero-free subset sums, Acta Arith. 78 (1996), no. 2, 143--152, DOI 10.4064/aa-78-2-143-152 (received 18 January 1996); Theorem 3.3 on printed p. 148 (PDF p. 6 of the publisher's ten-page file), read on the page image and in the text layer.

Read depth. Claims checked: the statement and the introduction's account (p. 143) of the constants $c=2$ for prime order (via Olson) and $c=3$ in general were read clause by clause. The half-page proof was read for structure only.

Proof pointer

If $0\notin\Sigma^*(S)$ then $S\cap(-S)=\emptyset$; with $s=|S|$ and $k=s-\lceil\log_{3/2}s\rceil$, Corollary 3.2 (a lower bound for $|\Sigma^*(S)|$ from the Cauchy--Davenport theorem and the isoperimetric connectivity $\kappa$) gives a contradiction once $\tfrac12k(k+1)-\tfrac94k(1+\ln k)\ge p$ (display (10)), which the stated hypothesis guarantees for $p\ge1000$; smaller $p$ are covered by Theorem 2.6 (p. 144, Olson's bound: $|S|>\sqrt{4p-3}$ implies $0\in\Sigma^*(S)$).

Dependencies

The Cauchy--Davenport theorem (Theorem 2.1), three theorems of Olson (Section 2) and the paper's Lemma 3.1 and Corollary 3.2.

Bears on

• Problem 540: for prime $N$ the threshold is $\sqrt{2N}$ up to $5\ln N$, close to the constant $\sqrt2$ Erdős speculated; superseded for primes by Balandraud's exact result and extended to all finite abelian groups by Theorem 4.5.