On zero sum subsequences of restricted size
Let \(G\) be a finite abelian group. Let \(s(G)\) be the minimal integer \(t\) with the property that any sequence of \(t\) elements in \(G\) contains a subsequence of length \(l=\exp(G)\) with sum zero. Similarly, let \(\eta(G)\) denote the corresponding minimal integer \(t\) where the subsequences are of size \(l \leq \exp(G)\). \textit{N. Alon} and \textit{M. Dubiner} [``A lattice point problem and additive number theory'', Combinatorica 15, No. 3, 301--309 (1995; Zbl 0838.11020)] proved that \(s(C_n^r) \leq c(r) n\) but the determination of \(s(C_n^r)\) (or \(\eta(C_n^r)\)) is a major open problem in this subject. For fixed \(n\) it is expected that \(\eta(C_n^r)=a_{r}(n-1)+1\) and \(s(C_n^r)=(a_{r}+1)(n-1)+1\) holds for some integer \(a_r\). The author proves: If \(\eta(C_m^r)=a_{r}(m-1)+1\) holds, and if \[ n \geq \frac{m^r( c(r)m -a_r(m-1)+m-3 )(m-1)-(m+1)+(m+1)(a_r+1)}{m (m+1)(a_r+1)} \] is a positive integer with \(s(G)=(a_r+1)(n-1)+1\), then \(s(C_{nm}^r)=(a_r+1)(nm-1)+1\).
scientific article; zbMATH DE number 5826102
scientific article