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

For sufficiently large subsets A,B, C,D of Fq, Gyarmati and Sárközy (2008) showed the solvability of the equations a+b = cd and ab+1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. They asked whether one can extend these results to every k ∈ N in the following way: for large subsets A,B, C,D of Fq, there are a1, . . . , ak, a1, . . . , ak ∈ A, b1, . . . , bk, b ′ 1, . . . , b ′ k ∈ B with ai + bj , aibj + 1 ∈ CD (for 1 6 i, j 6 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.