Is the set $\phi(\mathbb{N})$ syndetic?

Question

A set $A \subset \mathbb{N}$ is said to be syndetic if the gaps in $A$ are bounded.

Is the set $\phi(\mathbb{N})$ syndetic? (where $\phi$ denotes de Euler totient function)

I've thought quite a bit about this problem, but could not solve it and don't even have a clear idea of what the answer may be (mainly because it's hard to extract information about the values that the function $\phi$ in consecutive integers). Any help? Thanks!

Answer 1

The answer is negative. Assuming that $\phi(\mathbb{N})$ is a syndetic set we have that the totient numbers (i.e. the numbers that are of the form $\phi(n)$ for some $n$) have a positive lower asymptotic density, but Wikipedia contradicts that, giving us that the number of totient numbers up to $x$ is

$$ \frac{x}{\log x}\,\exp\left((C+o(1))\left(\log\log\log x\right)^2\right) $$ for some constant $C = 0.8178146\ldots$.

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That asymptotic is very interesting, because it shows that the problem is more or less equivalent to the trivial one:

Is the set of primes a syndetic set?

In such a case the answer is negative, since by considering primorials and by exploiting the PNT we have the existence of $k$ consecutive composite numbers less than $\exp((1+o(1))k)$.