On Modular $k$-Free Sets

Abstract

Let $n$ and $k$ be integers. A set $A\subset\mathbb{Z}/n\mathbb{Z}$ is $k$-free if for all $x$ in $A$, $kx\notin A$. We determine the maximal cardinality of such a set when $k$ and $n$ are coprime. We also study several particular cases and we propose an efficient algorithm for solving the general case. We finally give the asymptotic behaviour of the minimal size of a $k$-free set in $\left[ 1,n\right]$ which is maximal for inclusion.