Inverse problems associated with \(k\)-sums of sequences over finite abelian groups. (English) Zbl 1524.11066
Summary: Let \(G = C_{n_1}\oplus\cdots\oplus C_{n_r}\) be a finite abelian group with \(1 < n_1 \vert \cdots \vert n_r\) and \(S\) be a sequence over \(G\). Let \(\sum_k(S)\) denote the set of group elements which can be expressed as a sum of a subsequence of \(S\) with length \(k\). In this paper, we show that if \(0\ne \sum_{\vert G\vert}(S)\) and \(\vert S\vert = \vert G\vert +n_r -1+k\), where \(k \in [0,n_{r-1}-2]\), then \(\vert \sum_{\vert G\vert}(S)\vert \ge n_r(k+2)-1\) for some special cases. Moreover, we determine the structure of the sequence \(S\) when \(\vert \sum_{\vert G\vert}(S)\vert \) reaches the lower bound.
MSC:
| 11B75 | Other combinatorial number theory |
| 11P70 | Inverse problems of additive number theory, including sumsets |
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