On modular k-free sets

TL;DR

This paper investigates the maximum size of k-free sets in modular arithmetic, providing exact results for coprime cases, analyzing specific instances, and proposing an efficient algorithm for the general problem.

Contribution

It determines the maximal size of k-free sets in modular groups when k and n are coprime and introduces an efficient algorithm for the general case.

Findings

Maximal size of k-free sets for coprime n and k determined

Efficient algorithm proposed for general k-free set problem

Asymptotic behavior of minimal size of maximal k-free sets analyzed

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.