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The generalized k-resultant modulus set problem in finite fields

Research paper by David Covert, Doowon Koh, Youngjin Pi

Indexed on: 01 Mar '17Published on: 01 Mar '17Published in: arXiv - Mathematics - Combinatorics



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 \[ \Delta_k(E_1,E_2, \ldots, E_k)=\left\{\|x^1+x^2+\cdots+x^k\|\in \mathbb F_q:x^j\in E_j,\, j=1,2,\ldots, k\right\},\] where $\|y\|=y_1^2+ \cdots + y_d^2$ for $y=(y_1, \ldots, y_d)\in \mathbb F_q^d.$ We prove that if $\prod\limits_{j=1}^3 |E_j| \ge C q^{3\left(\frac{d+1}{2} -\frac{1}{6d+2}\right)}$ for $d=4,6$ with a sufficiently large constant $C>0$, then $|\Delta_3(E_1,E_2,E_3)|\ge cq$ for some constant $0<c\le 1,$ and if $\prod\limits_{j=1}^4 |E_j| \ge C q^{4\left(\frac{d+1}{2} -\frac{1}{6d+2}\right)}$ for even $d\ge 8,$ then $|\Delta_4(E_1,E_2,E_3, E_4)|\ge cq.$ We also show that if $\prod\limits_{j=1}^3 |E_j| \ge C q^{3\left(\frac{d+1}{2} -\frac{1}{9d-18}\right)}$ for even $d\ge 8,$ then $|\Delta_3(E_1,E_2,E_3)|\ge cq.$ This work both generalizes and improves previous work by the authors.