On the sums of any k points in finite fields

Abstract

For a set $E\subset {\mathbb{F}}_q^d$, we define the $k$-resultant magnitude set as ${\Delta }_k(E)=\{\Vert {\text{x}}_1+\cdots +{\text{x}}_k\Vert \in {\mathbb{F}}_q:{\text{x}}_1,\dots ,{\text{x}}_k\in E\},$ where $\Vert \text{v}\Vert =v_1^2+\cdots +v_d^2$ for $\text{v}=(v_1,\dots ,v_d)\in {\mathbb{F}}_q^d.$ 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 $|E|\ge C{q}^{\frac{d+1}{2}-\frac{1}{6d+2}},$ then $|{\Delta }_3(E)|\ge cq$ for $d=4$ or $d=6$, and $|{\Delta }_4(E)|\ge cq$ for even dimensions $d\ge 8.$ In addition, we prove that if $d\ge 8$ is even, and $|E|\ge C_{\epsilon }{q}^{\frac{d+1}{2}-\frac{1}{9d-18}+\epsilon }$ for $\epsilon >0$, then $|{\Delta }_3(E)|\ge cq.$

16 pages

References in corpus (2)

• Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields
• Distance sets of two subsets of vector spaces over finite fields