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Davenport constant with weights and some related questions. (English) Zbl 1107.11018

Let \(n\in\mathbb N\) and \(A\subseteq\mathbb Z/n\mathbb Z\) \((0\not\in A\neq\emptyset)\). The number \(E_A(n)\) is the least \(t\in\mathbb N\) such that for all sequences \((x_1,\dots x_t)\in\mathbb Z^t\), there exist indices \(j_1,\dots, j_n\in\mathbb N\) \((1\leq j_1<\cdots< j_n\leq t)\) and \((\vartheta_1,\dots, \vartheta_n)\in A^n\) with \(\sum^n_{i=1} \vartheta_i x_{j_k}\equiv 0\pmod n\). Similarly, for any such set \(A\) is defined the Davenport constant \(D_A(n)\) of \(\mathbb Z/n\mathbb Z\) with weight \(A\) the least \(k\in\mathbb N\) such that for any sequence \((x_1,\dots, x_k)\in\mathbb Z^k\), there exists a non-empty subsequence \(\{x_{j_1},\dots, x_{j_\ell}\}\) and \((a_1,\dots, a_\ell)\in A^\ell\) such that \(\sum^\ell_{i=1} a_i x_{j_i}\equiv 0\pmod n\).
In this paper for a prime \(p\) are proved the following two theorems:
(1) Let \(A= \{1,2,\dots, r\}\) with \(1< r< p\). Then (i) \(D_A(p)= \lceil{p\over r}\rceil\); (ii) \(E_A(p)= p- 1+ D_A(p)\).
(2) Let \(A\) be the set of quadratic residues mod \(p\). Then (i) \(D_A(p)= 3\); (ii) \(E_A(p)- p+ 2\).

MSC:

11B75 Other combinatorial number theory
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