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Statement

Theorem 4.5 (p. 151). Some function $\varepsilon(n)=O(n^{1/3}\ln n)$ has the following property. Let $G$ be a finite abelian group of order $n$ and let $S\subset G$. Then

$$|S|>\sqrt{2n}+\varepsilon(n)\quad\text{implies}\quad0\in\Sigma^*(S),$$

where $\Sigma^*(S)$ is the set of sums of nonempty subsets of $S$.

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; Theorem 4.5 on printed p. 151 (PDF p. 9), read on the page image and in the text layer. The inequality is printed strict ($>$), although the introduction (p. 143) states the result with $\ge$.

Read depth. Claims checked: the statement was read clause by clause. The proof (Section 4, pp. 148--151) was read for structure only.

Proof pointer

Section 4 adapts the prime-order argument: Lemma 4.1 extracts from $S$ a subset $K$ with $K\cap(-K)=\emptyset$ and bounds $\kappa(S\cup(-S))$; Lemma 4.3 and Corollary 4.4 give $|\Sigma^*(T)|\ge\min(n,\tfrac12k(k+1)-3(k+1)^{3/2}-3dk(1+\ln k))$ (display (13)) for a subset $T$ of $S$ under a hypothesis on the subgroups generated by large subsets of $S$, which Theorem 2.5 supplies when $|S|>3\sqrt{n/d}+k+2d\log_{3/2}n$ (display (14)); choosing $d\sim n^{1/3}$ gives the theorem.

Dependencies

Kneser's and Scherk's theorems, Olson's Theorem 2.5 and the paper's Section 4 lemmas.

Bears on

• Problem 540: the best general bound in hand, the site's "Hamidoune and Zémor proved the bound $(1+o(1))\sqrt{2N}$ for arbitrary abelian groups of order $N$"; it sharpens Szemerédi's unspecified constant to $\sqrt2$ up to a lower-order term.