On Sum of Products and Erd\H{o}s Distance Problem over Finite Fields

For a prime power $q$, let $\mathbb{F}_q$ be the finite field of $q$ elements. We show that \mathbb{F}_q^{\ast} \subseteq d\mathcal{A}^2$ for almost every subset $\mathcal{A} \subset \mathbb{F}_q$ of cardinality $|\mathcal{A}| \gg q^{1 / d}$. Furthermore, if $q = p$ is a prime, and $\mathcal{A} \subseteq \mathbb{F}_p$ of cardinality $|\mathcal{A}| \gg p^{1 / 2} (\log p)^{1/d}$, then $d\mathcal{A}^2$ contains both large and small residues. We also obtain some results of this type for the Erd\H{o}s distance problem over finite fields