Zero-sum constants involving weights
Let \(G\) be a finite abelian group written additively with exponent \(\exp (G)=m.\) The \textit{Davenport constant} of \(G\), denoted by \(D(G),\) is the least positive integer \(k\) such that, for any sequence \((x_1,\dots,x_k)\) with \(x_j\in G, 1\leq j\leq k,\) there exists a non-empty subsequence \((x_{j_1}, \dots, x_{j_\ell})\) with \(1\leq j_1<\dots<j_\ell\leq k\) having zero sum: \(\sum_{i=1}^\ell x_{j_i}=0.\) If \(G={ Z}_n,\) the cyclic group of the the residues classes modulo \(n\), then \(D(G)=n.\) Consider now as \textit{set of weights} a non-empty subet \(A\) of \(\{1,\dots,m-1\}.\) The \textit{weighted Davenport constant} of \(G\) with weights in \(A\), denoted by \(D_A(G),\) is the least positive integer \(k\) such that, for any sequence \((x_1,\dots,x_k)\) with \(x_j\in G, 1\leq j\leq k,\) there exists a non-empty subsequence \((x_{j_1}, \dots, x_{j_\ell})\) with \(1\leq j_1<\dots<j_\ell\leq k\) and elements \(a_1,\dots,a_{\ell}\) (repetitions allowed) of \(A\) such that \(\sum_{i=1}^\ell a_i x_{j_i}=0.\) If \(A=\{1\},\) then \(D_A(G)\) is \(D(A).\) In the introduction, the authors present a precise and useful collection of results on \(D_A(Z_n)\) for various choices of the weight set \(A.\) The term ``constants'' in the title is in plural because, in some cases, the relation between \(D_A(G)\) and the so called Gao (or Erdős) constant \(E_A(G)\) is exhibited. The authors prove the next three results. Theorem 4. Let \(n\) and \(d\) be positive integers with \(1<d\leq n.\) If \(A=\{a,a+1,\dots,a+\lceil n/d \rceil -1\}\subset\{1,2,\dots,n-1\},\) then \(D_A(Z_n)\leq d.\) Theorem 5. If \(n=p_1^{a_1}\cdots p_k^{a_k},\) where the \(p_i\)'s are distinct primes, and \(1\leq s\leq k,\) then taking \(A_s=\{a:1\leq a\leq n-1,(a,p_1^{a_1}\cdots p_s^{a_s})=1\},\) we have \(D_{A_s}(Z_n)=a_1+\cdots+a_s+1.\) Theorem 6. Let \(n\) be a positive integer and \(k\) be an integer such that \(1\leq k\leq \lfloor n/2 \rfloor \). Let \(A=\{\pm 1,\pm 2,\dots,\pm k\}.\) Then we have \(D_{A}(Z_n)=\lfloor\log_{(k+1)}n \rfloor +1.\) In the proofs, the authors use a lemma from [\textit{P. Scherk} and \textit{J. H. B. Kemperman}, Can. J. Math. 6, 230--237 (1954; Zbl 0058.01804)].
scientific article; zbMATH DE number 7434834
scientific article