Energy of sets on a paraboloid in finite fields

Let $mathbb{F}_q$ be a finite field of order $q$, and $P$ be the paraboloid in $mathbb{F}_q^3$ defined by the equation $z=x^2+y^2$. A tuple $(a, b, c, d)in P^4$ is called a non-trivial energy tuple if $a+b=c+d$ and $a, b, c, d$ are distinct. For $Xsubset P$, let $mathcal{E}^+(X)$ be the number of non-trivial energy tuples in $X$. It was proved recently by Lewko (2020) that $mathcal{E}^+(X)ll |X|^{frac{99}{41}}$ for $|X|ll q^{frac{26}{21}}$. The main purposes of this paper are to prove lower bounds of $mathcal{E}^+(X)$ and to study related questions by using combinatorial arguments and a weak hypergraph regularity lemma developed recently by Lyall and Magyar (2020).