Pseudo-random subsets constructed by using Fermat quotients

Let $p$ be a prime, and let $n$ be an arbitrary integer with $(n,p)=1$. The Fermat quotient $q_p(n)$ is defined as the unique integer with $$ q_p(n)\equiv\frac{n^{p-1}-1}{p} \ (\bmod\ p),\qquad 0\leq q_p(n)\leq p-1. $$ We also define $q_p(kp)=0$ for $k\in\mathbb{Z}$. In this paper, we study the pseudo-randomness of subsets constructed by Fermat quotients, by using the estimates for exponential sums and character sums with Fermat quotients.