The generalized k-resultant modulus set problem in finite fields

Let ${\mathbb{F}}_q^d$ be the $d$-dimensional vector space over the finite field ${\mathbb{F}}_q$ with $q$ elements. Given $k$ sets $E_j\subset {\mathbb{F}}_q^d$ for $j=1,2,\dots ,k$, the generalized $k$-resultant modulus set, denoted by ${\mathrm{\Delta }}_k(E_1,E_2,\dots ,E_k)$, is defined by $${\mathrm{\Delta }}_k(E_1,E_2,\dots ,E_k)=\{\mathrm{\parallel }{\mathbf{x}}^1+{\mathbf{x}}^2+\cdots +{\mathbf{x}}^k\mathrm{\parallel }\in {\mathbb{F}}_q:{\mathbf{x}}^j\in E_j,j=1,2,\dots ,k\},$$ where $|{\bf y}|={\bf y}_1^2+ \cdots + {\bf y}d^2$for${\bf y}=({\bf y}1, \ldots, {\bf y}d)\in \mathbb F_q^d.$Weprovethatif$\prod\limits{j=1}^3 |E_j| \ge C q^{3\left(\frac{d+1}{2} -\frac{1}{6d+2}\right)}$for$d=4,6$ with a sufficiently large constant $C>0$, then $\mathrm{\mid }{\mathrm{\Delta }}_3(E_1,E_2,E_3)\mathrm{\mid }\ge cq$ for some constant $0<c\le 1,$ and if $\prod\limits{j=1}^4 |E_j| \ge C q^{4\left(\frac{d+1}{2} -\frac{1}{6d+2}\right)}$foreven$d\ge 8,$then$|\Delta_4(E_1,E_2,E_3, E_4)|\ge cq.$ This generalizes the previous result in \cite{CKP16}. We also show that if $\prod\limits{j=1}^3 |E_j| \ge C q^{3\left(\frac{d+1}{2} -\frac{1}{9d-18}\right)}$foreven$d\ge 8,$then$|\Delta_3(E_1,E_2,E_3)|\ge cq.$ This result improves the previous work in \cite{CKP16} by removing $\epsilon >0$ from the exponent.