Abstract
Let \(G=C_{n_1}\oplus \cdots \oplus C_{n_r}\) be a finite abelian group with \(1 < n_1 \mid \cdots \mid n_r\) and S be a sequence over G. Let \(\Sigma_k(S)\) 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 \(0\notin \Sigma_{|G|}(S)\) and \(|S|=|G|+n_r-1+k\), where \(k\in[0, n_{r-1} -2]\), then \(|\Sigma_{|G|}(S)|\geq n_r(k+2)-1\) for some special cases. Moreover, we determine the structure of the sequence S when \(|\Sigma_{|G|}(S)|\) reaches the lower bound.
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Acknowledgements
We would like to thank the referee for valuable suggestions which helped improve the presentation of the paper. A part of this work was carried out during a visit by the first author to Brock University as an international visiting scholar. She would like to sincerely thank the host institution for its hospitality and for providing an excellent atmosphere for research.
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This work was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN2017-03903) and the National Natural Science Foundation of China (Grant No. 12271520).
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Hui, W., Li, Y., Peng, J. et al. Inverse problems associated with k-sums of sequences over finite abelian groups. Acta Math. Hungar. 169, 312–324 (2023). https://doi.org/10.1007/s10474-023-01303-z
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DOI: https://doi.org/10.1007/s10474-023-01303-z