### Sum and Shifted-Product Subsets of Product-Sets over Finite Rings

#### Abstract

For sufficiently large subsets $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}$ of $\mathbb{F}_q$, Gyarmati and Sárközy (2008) 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 $a_1, \ldots, a_k, a_1', \ldots, a_k' \in\mathcal{A}$, $b_1, \ldots, b_k, b_1', \ldots, b_k' \in \mathcal{B}$ with $a_i + b_j, a_i' b_j' + 1 \in \mathcal{C}\mathcal{D}$ (for $1 \leq i, j\leq k)$. The author (2010) 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.