The Generalized k-Resultant Modulus Set Problem in Finite Fields

Abstract

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 $j=1,2,\ldots , k$, the generalized k-resultant modulus set, denoted by $\Delta _k(E_1,E_2, \ldots , E_k)$, is defined by $\Delta _k(E_1,E_2, \ldots , E_k)=\{\Vert \mathbf{x}^1+\mathbf{x}^2+\cdots +\mathbf{x}^k\Vert \in {\mathbb {F}}_q:\mathbf{x}^j\in E_j,\, j=1,2,\ldots , k\},$ where $\Vert \mathbf{y}\Vert =\mathbf{y}_1^2+ \cdots + \mathbf{y}_d^2$ for $\mathbf{y}=(\mathbf{y}_1, \ldots , \mathbf{y}_d)\in {\mathbb {F}}_q^d.$ We prove that if $\prod \nolimits _{j=1}^3 |E_j| \ge C q^{3\left( \frac{d+1}{2} -\frac{1}{6d+2}\right) }$ for $d=4,6$ with a sufficiently large constant $C>0$, then $|\Delta _3(E_1,E_2,E_3)|\ge cq$ for some constant $0<c\le 1,$ and if $\prod \nolimits _{j=1}^4 |E_j| \ge C q^{4\left( \frac{d+1}{2} -\frac{1}{6d+2}\right) }$ for even $d\ge 8,$ then $|\Delta _4(E_1,E_2,E_3, E_4)|\ge cq.$ This generalizes the previous result in [3]. We also show that if $\prod \nolimits _{j=1}^3 |E_j| \ge C q^{3\left( \frac{d+1}{2} -\frac{1}{9d-18}\right) }$ for even $d\ge 8,$ then $|\Delta _3(E_1,E_2,E_3)|\ge cq.$ This result improves the previous work in [3] by removing $\varepsilon >0$ from the exponent. The new ingredient in our proof is an improved $L^3$-restriction estimate for spheres.