Long zero-free sequences in finite cyclic groups

Abstract

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 $\overline{g{a}_i}$ denotes the least positive integer in the congruence class $g{a}_i$ (modulo $n$), then ${\Sigma }_{i=1}^{\ell }\overline{g{a}_i}<n$. 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.

13 pages