In this paper, we apply character sum estimates to study products of factorials modulo $p$.

Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$. In other words, there exists a non-trivial character $\chi \in \widehat{G}$ such that ${\sum }_{a\in A}\chi \left(a\right)\in P$.

We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$. We also establish an asymptotic formula for $1\le x,y,z\le H$ such that $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$ are square-free. The method we used in this paper is due to Tolev.