A Generalisation of Maximal $(k,b)$-Linear-Free Sets of Integers

Abstract

Fix integers $k,b,q$ with $k\ge 2,b\ge 0,q\ge 2$. Define the function $p$ to be: $p(x)=kx+b$. We call a set $S$ of integers $(k,b,q)$-linear-free if $x\in S$ implies $p^i (x)\notin S$ for all $i=1,2,\dots,q-1$, where $p^i (x)=p\left(p^{i-1}(x)\right)$ and $p^0 (x)=x$. Such a set $S$ is maximal in $[n]:=\{1,2,\dots,n\}$ if $S\cup \{t\},\forall t\in [n]\backslash S$ is not $(k,b,q)$-linear-free. Let $M_{k,b,q} (n)$ be the set of all maximal $(k,b,q)$-linear-free subsets of $[n]$, and define $g_{k,b,q} (n)=\min_{S\in M_{k,b,q} (n)}|S|$ and $f_{k,b,q} (n)=\max_{S\in M_{k,b,q} (n)}|S|$. In this paper, formulae for $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$ are proposed. Also, it is proven that there is at least one maximal $(k,b,q)$-linear-free subset of $[n]$ with exactly $x$ elements for any integer $x$ between $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$, inclusively.