Density properties of fractions with Euler's totient function

We prove that for all constants $a\in \mathbb{N}$, $b\in \mathbb{Z}$, $c,d\in \mathbb{R}$, $c\mathrm{\ne }0$, the fractions $\phi (an+b)\mathrm{/}(cn+d)$ lie dense in the interval $]0,D]$ (respectively $[D,0[$ if $c<0$), where $D=a\phi (\gcd (a,b))\mathrm{/}(c\gcd (a,b))$. This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the amount of $n$ such that $\text{rad}(an+b)\mathrm{\mid }g$ for coprime $a,b$ and any $g$. Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V.~Arnold. For the fractions $\phi (an+b)\mathrm{/}\phi (cn+d)$ with constants $a,c\in \mathbb{N},b,d\in \mathbb{Z}$, we prove that they lie dense in $]0,\mathrm{\infty }[$ exactly if $ad\mathrm{\ne }bc$.