Density and ramsey type results on algebraic equations with restricted solution sets

Abstract

In earlier papers Sarkozy studied the solvability of the equations $$a + b = cd, a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D},$$ resp. $$ab + 1 = cd, a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D}$$ where \(\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\) are “large” subsets of \(\mathbb{F}_p\). Later Gyarmati and Sarkozy generalized and extended these problems by studying these equations and also other algebraic equations with restricted solution sets over finite fields. Here we will continue the work by studying further special equations over finite fields and also algebraic equations with restricted solution sets over the set of the integers, resp. rationals. We will focus on the most interesting cases of algebraic equations with 3, resp. 4 variables. In the cases when there are no “density results” of the above type, we will be also looking for Ramsey type results, i.e., for monochromatic solutions of the given equation. While in the earlier papers character sum estimates were used, now combinatorial tools dominate.