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

Keywords:

Davenport constant
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