06045726

j

06045726 Vinh, Le Anh Sum and shifted-product subsets of product-sets over finite rings. Electron. J. Comb. 19, No. 2, Research Paper P33, 9 p., electronic only (2012). 2012 Prof. Andr\'e K\"undgen, Deptartment of Mathematics, California State University San Marcos, San Marcos, CA EN graph theory sum-product sets residue rings Zbl 05982464

emis:journals/EJC/ojs/index.php/eljc/article/view/v19i2p33

Summary: For sufficiently large subsets $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}$ of ${\mathbb{F}}_q$, {\it K. Gyarmati} and {\it A. S\'ark\"ozy} [Acta Math. Hung. 118, No. 1--2, 129--148 (2008; Zbl 1164.11074) and ibid. 119, No. 3, 259--280 (2008; Zbl 1199.11141)] showed the solvability of the equations $a + b= c d$ and $a b + 1 = c d$ with $a \in \mathcal{A}$, $b \in \mathcal{B}$, $c \in \mathcal{C}$, $d \in \mathcal{D}$. They asked whether one can extend these results to every $k \in \mathbb{N}$ in the following way: for large subsets $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}$ of ${\mathbb{F}}_q$, there are $a_1, \dots, a_k, a_1', \dots, a_k' \in \mathcal{A}$, $b_1, \dots,b_k, b_1', \dots,b_k' \in \mathcal{B}$ with $a_i + b_j, a_i' b_j' + 1 \in \mathcal{C}\mathcal{D}$ (for $1 \leq i$, $j\leq k$). The author [Eur. J. Comb. 32, No. 8, 1177--1181 (2011; Zbl 05982464)] gave an affirmative answer to this question using Fourier analytic methods. In this paper, we will extend this result to the setting of finite cyclic rings using tools from spectral graph theory.