Remarks on a generalization of the Davenport constant. (English) Zbl 1228.05302
Summary: A generalization of the Davenport constant is investigated. For a finite abelian group \(G\) and a positive integer \(k\), let \(D_k(G)\) denote the smallest \(\ell \) such that each sequence over \(G\) of length at least \(\ell \) has \(k\) disjoint non-empty zero-sum subsequences. For general \(G\), expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence \(D_k(G)\) is eventually an arithmetic progression with difference exp\((G)\), and several questions arising from this fact are investigated. For elementary 2-groups, \(D_k(G)\) is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).
Keywords:
Davenport constant; non-unique factorization; set of lengths; zero-sum sequence; Krull monoidReferences:
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