On the maximum cardinality of a $k$-zero-free set in an Abelian group

A subset $A$ of elements of an Abelian group $G$ is called $k$-zero-free if $x_1+\dots+x_{k-1}$ does not belong to $A$ for any $x_1,\dots,x_{k-1}\in A$. A $k$-zero-free set $A$ in the group $G$ is called maximal if for any $x\in G\setminus A$ the set $A\cup\{x\}$ is not $k$-zero-free. We study the maximum cardinality of a $k$-zero-free set in an Abelian group $G$. In particular, the maximum cardinality of a $k$-zero-free arithmetic progression in a cyclic group $Z_n$ is determined and upper and lower bounds on the maximum cardinality of a $k$-zero-free set in an Abelian group $G$ are improved. We describe the structure of $k$-zero-free maximal sets $A$ in the cyclic group $Z_n$ if $\mathrm{gcd}(n,k)=1$ and $k|A|\ge n+1$. Bibliogr. 8.