Let $\mathbb P$ be the set of prime numbers. 


Is there a non constant polynomial $f \in \mathbb Z[X]$ such that the set $$I\_f := \{ \textstyle\frac{z}{p} : z \in \mathbb Z, p \in \mathbb P, p \mid f(z) \}$$
is dense in $\mathbb R$?


(I suppose that the following much stronger statement holds: Every polynomial $f \in \mathbb Z[X]$ which has a irreducible factor of degree at least $2$ satisfies the above condition.)



Distribution of zeros of a polynomial mod. a prime
The set $I\_f$ is dense for quadratic polynomials which are irreducible over $\mathbb{Q}$, by the work of Duke-Friedlander-Iwaniec (Ann. of Math. 141 (1995), 423-441) and Toth (IMRN 2000, No. 14, 719-739). In fact we have uniform distribution even when $p$ is restricted to a reduced residue class of any modulus.


