Local behaviour of the set of the values of Euler's totient function

Let $V$ be the set of the values of the Euler's $\varphi$ function. It is known that the natural density of $V$, namely the limit of $V(x)/x$, as $x$ tends to infinity, is $0$.

In a paper to appear, M. K. Das, P. Eyyunni and B. R. Patil claim that this also applies to a local density of $V$, known as the uniform upper density, or the Banach density, defined by
$$
\delta^{*(}V) = \lim_{H \rightarrow \infty} \limsup_{x \rightarrow \infty} \frac{1}{H}\left(V(x+H) -V(x)\right).
$$
The aim of this lecture is to show that, under two standard conjectures (Schinzel's $H$-hypothesis and Masser-Oesrtl\'{e}'s $abc$-conjecture), the Banach density of $V$ is $1/4$, a joint work with P. Eyyunni and S. Gun.

Getting around the use of the $abc$-conjecture raises an interesting Diophantine problem.