# 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
$$C(n,d)geq frac{p-1}{b(n,d)n}quad?$$