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Davenport constant with weights and some related questions. II. (English) Zbl 1210.11031

Summary: Let \(G\) be a finite abelian group of order \(n\) and \(A\subseteq\mathbb Z\) be non-empty. Generalizing a well-known constant, we define the Davenport constant of \(G\) with weight \(A\), denoted by \(D_A(G)\), to be the least natural number \(k\) such that for any sequence \((x_1,\dots, x_k)\) with \(x\in G\), there exists a non-empty subsequence \((x_{j_1},\dots,x_{j_l})\) and \(a_1,\dots,a_l\in A\) such that \(\sum_{i=1}^la_ix_{j_i}=0\). Similarly, for any such set \(A\), \(E_A(G)\) is defined to be the least \(t\in \mathbb N\) such that for all sequences \((x_1,\dots,x_t)\) with \(x_i\in G\), there exist indices \(j_1,\dots,j_n\in \mathbb N\), \(1\leq j_1<\dots <j_n\leq t\) and \(\vartheta_1,\dots,\vartheta_n\in A\) with \(\sum_{i=1}^n\vartheta_ix_{j_i}=0\). In the present paper, we establish a relation between the constants \(D_A(G)\) and \(E_A(G)\) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group \(G=\mathbb Z/n\mathbb Z\) and the relation we establish had been conjectured in that particular case.
Part I, cf. the first author and P. Rath, Integers 6, Paper A30, 6 p., electronic only (2006; Zbl 1107.11018).

MSC:

11B75 Other combinatorial number theory
05D10 Ramsey theory
20K01 Finite abelian groups

Citations:

Zbl 1107.11018
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References:

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