nt.number theory – Is there a way to gain such a estimate?

This problem could be viewed as a polynomial generalization of the Lonely runner conjecture. Take $nin mathbb{N}^*$ fixed, $A_p subset (mathbb{Z} / p mathbb{Z})^{times}$ is a finite set, pur $|A_p|=n$. $fin mathbb{Z}(X)$ is a polynomial, assume that it satisfies property $P$ for $pinBbb N^ast$, i.e.
$$newcommand{degr}{operatorname{deg}}
f(x)neq f(y) quad forall xneq y,;x,yin mathbb{Z} / p mathbb{Z},
$$

put $degr(f)=din mathbb{N}^*$ and finally consider
$$C(n,d)=liminf_{substack{p text{ prime,} \ ftext{ satisfies property $P$ for $p$}}}min_{substack{fin mathbb{Z}(X),\degr(f)=din mathbb{N}^*}} max
_{tin mathbb{Z} / p mathbb{Z}}min_{a,bin A_p}|f(ta)-f(tb)|.label{1}tag{1}
$$

We can prove a trivial estimate on eqref{1} by using a “volume” counting argument, i.e.
$$C(n,d)geq frac{p-1}{2n}$$
Question: in general, can we improve the constant $2$, i.e. do there exists a constant $b(n,d)<2$, such that
$$
C(n,d)geq frac{p-1}{b(n,d)n}quad?
$$