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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