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On a conjecture of Hamidoune for subsequence sums. (English) Zbl 1098.11019

Let \((G,+,0)\) be an abelian group. For \(A,B\subseteq G\) is the sumset \(A+B:= \{a+b\mid a\in A,b\in B\}\). For a group \(G\) of order \(m\) and for a sequence \(S\) with \(k\) distinct terms from \(G\) let \(m\wedge S\) denote the set of all elements that are a sum of some \(m\)-term subsequence of \(S\), and let \(|S|\) be the length of \(S\).
Here the following result is shown (Theorem 1): If \(|S|\geq m+1\), and if the multiplicity of each term of \(S\) is at most \(m-k+2\), then either \(|m\wedge S|\geq \min\{m,|S|-m+k-1\}\) or there exists a proper, nontrivial subgroup \(H_a\) of index \(a\), such that \(m\wedge S\) is a union of \(H_a\)-cosets, \(H_a\subseteq m\wedge S\), and all but \(e\) terms of \(S\) are from the same \(H_a\)-coset, where \(e\leq\{\lfloor\frac{|S|-m+k-2}{|H_a|}\rfloor-1,a-2\}\) and \(|m\wedge S|(e+1) |H_a|\).
This confirms a conjecture of Y. O. Hamidoune [Comb. Probab. Comput. 12, 413–425 (2003; Zbl 1049.11024)].
Theorem 1 also implies that if \(|m\wedge S|<|S|-m+k-1\), then \(m\wedge S\) is nontrivial periodic, a conclusion similar to the classical result of M. Kneser for sumsets [Math. Z. 58, 459–484 (1953; Zbl 0051.28104); Math. Z. 61, 429–434 (1955; Zbl 0064.04305)].

MSC:

11B83 Special sequences and polynomials
20K01 Finite abelian groups
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