On the sums of any k points in finite fields

TL;DR

This paper investigates the size of the sumsets of points in finite fields, establishing lower bounds based on the set size and connecting these bounds to restriction theorems for spheres.

Contribution

It introduces new bounds for the cardinality of sumsets in finite fields and links these bounds to restriction theorems, advancing understanding of sumset behavior in finite field geometry.

Findings

For certain dimensions, large enough sets ensure the sumset covers a positive proportion of the field.

Established bounds depend on the dimension and size of the set.

Connected sumset size to restriction theorems for spheres in finite fields.

Abstract

For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \Delta_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 $|\Delta_3(E)|\geq c q$ for $d = 4$ or $d = 6$, and $|\Delta_4(E)| \geq cq$ 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 $|\Delta_3(E)|\geq c q.$