On the sums of any k points in finite fields

For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $Δ_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$ for $\textbf{v}=(v_1, \ldots, v_d) \in \mathbb F_q^d.$ In this paper we find a connection between a lower bound of the cardinality of the $k$-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if $E\subset \mathbb F_q^d$ with $|E|\geq C q^{\frac{d+1}{2}-\frac{1}{6d+2}},$ then $|Δ_3(E)|\geq c q$ for $d = 4$ or $d = 6$, and $|Δ_4(E)| \geq cq$ for even dimensions $d \geq 8.$ In addition, we prove that if $d\geq 8$ is even, and $|E|\geq C_\varepsilon ~q^{\frac{d+1}{2} - \frac{1}{9d -18} + \varepsilon}$ for $\varepsilon >0$, then $|Δ_3(E)|\geq c q.$

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