Long zero-free sequences in finite cyclic groups
| dc.creator | Savchev, Svetoslav | |
| dc.creator | Chen, Fang | |
| dc.date | 2006-02-25 | |
| dc.date.accessioned | 2026-07-07T07:03:46Z | |
| dc.date.available | 2026-07-07T07:03:46Z | |
| dc.description | A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than $n/2$ in the additive group $\Zn/$ of integers modulo $n$. The main result states that for each zero-free sequence $(a_i)_{i=1}^\ell$ of length $\ell>n/2$ in $\Zn/$ there is an integer $g$ coprime to $n$ such that if $\bar{ga_i}$ denotes the least positive integer in the congruence class $ga_i$ (modulo $n$), then $Σ_{i=1}^\ell\bar{ga_i}<n$. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than $n/2$, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts. | |
| dc.description | 13 pages | |
| dc.identifier | https://arxiv.org/abs/math/0602568 | |
| dc.identifier | http://arxiv.org/abs/math/0602568 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/109104 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 11B50, 11P21 | |
| dc.title | Long zero-free sequences in finite cyclic groups | |
| dc.type | text |