On Multiplicative Sidon Sets

Fix integers $b>a\ge 1$ with $g:=\gcd (a,b)$. A set $S\subseteq \mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\mathrm{\ne }by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and conclude that the maximum density of an $\{a,b\}$-multiplicative set is $\frac{b}{b+g}$. For $A,B\subseteq \mathbb{N}$, a set $S\subseteq \mathbb{N}$ is \emph{$\{A,B\}$-multiplicative} if $ax=by$ implies $a=b$ and $x=y$ for all $a\in A$ and $b\in B$, and $x,y\in S$. For $1<a<b<c$ and $a,b,c$ coprime, we give an O(1) time algorithm to approximate the maximum density of an $\{\{a\},\{b,c\}\}$-multiplicative set to arbitrary given precision.