Hamidoune 1996 zero free subset sums

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theorem_3_3: The prime-order threshold √(2p) up to a logarithmic term.

theorem_4_5: The threshold √(2n) up to a lower-order term for arbitrary finite abelian groups.

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Hamidoune, Yahya Ould and Zémor, Gilles, On zero-free subset sums. Acta Arith. 78 (1996), no. 2, 143--152. The file's text layer carries no copyright or license line; the journal's record offers the PDF under the download link "Pobierz zgodnie z CC-BY", rendered "Free download under CC-BY license" on the English site, and names no version or URL for it (https://www.impan.pl/get/doi/10.4064/aa-78-2-143-152, read 2026-10-02): the Creative Commons Attribution license, with no version stated.

The paper attacks the Erdos-Heilbronn problem of showing that some constant c makes any subset S of a finite abelian group G with |S| >= c|G|^{1/2} contain a nonempty zero-sum subset; Erdos and Heilbronn conjectured c = 2, Erdos later stipulated c = sqrt2, and previous bounds were c = 2 for prime order (via Olson) and c = 3 in general. Theorem 3.3 proves that when G has prime order, any S with |S| >= sqrt2 |G|^{1/2} + 5 ln|G| has 0 in Sigma*(S), and Theorem 4.5 proves the same for arbitrary finite abelian G with |S| > sqrt2 |G|^{1/2} + epsilon(|G|) where epsilon(n) = O(n^{1/3} ln n). Thus the constant is reduced to sqrt2 up to lower-order terms, essentially settling the conjectured constant. The tools are classical addition theorems (Cauchy-Davenport, Kneser, Scherk), the connectivity invariant kappa(X) = min_M |(X+M)\M|, three subset-sum theorems of Olson, and an averaging technique introduced by Erdos and Heilbronn and developed by Olson. These bounds bear on problem 540 on zero-sum subsets of large subsets of abelian groups.

The retained folder-name PDF is the publisher's ten-page file (Acta Arith. 78 (1996), no. 2, 143--152, DOI 10.4064/aa-78-2-143-152; received 18 January 1996); printed p. n is PDF p. n - 142. Read status: claims checked for the introduction's history (p. 143), Theorem 3.3 (p. 148) and Theorem 4.5 (p. 151), read on the page images and in the text layer; the proofs were read for structure only. Theorem 4.5 is printed with a strict inequality, $|S|>\sqrt{2n}+\varepsilon(n)$. Result pages: theorem_3_3 and theorem_4_5.

Source: http://www.impan.pl/get/doi/10.4064/aa-78-2-143-152.

Bears on. #540

Results to transcribe.

• Theorem 3.3: For G of prime order, any S with |S| >= sqrt2 |G|^{1/2} + 5 ln|G| contains a nonempty subset summing to zero (p. 148; result page theorem_3_3).
• Theorem 4.5: For arbitrary finite abelian G, |S| > sqrt2 |G|^{1/2} + epsilon(|G|) with epsilon(n)=O(n^{1/3} ln n) forces a zero-sum subset (p. 151; result page theorem_4_5).
• Method: Cauchy-Davenport, Kneser and Scherk addition theorems plus the isoperimetric invariant kappa(X) and the Erdos-Heilbronn averaging argument as developed by Olson.