Equations in finite fields with restricted solution sets. II: Algebraic equations
Let \(\mathbb F_{q}\) be a finite field with \(q\) elements, and \(\mathcal A\), \(\mathcal B\), \(\mathcal C\), \(\mathcal D\) ``large'' subsets of \(\mathbb F_{q}\). Among several other results the authors prove that if \(| \mathcal A| |\mathcal B| |\mathcal C| |\mathcal D| > q^3\) (resp. \(| \mathcal A| |\mathcal B| |\mathcal C| |\mathcal D| > 100 q^3\)) then the equation \(a+b=cd\) (resp. the equation \(ab+1=cd\)) is solvable in elements \(a \in \mathcal A\), \(b \in\mathcal B\), \(c \in\mathcal C\) and \(d \in\mathcal D\). The proofs are based on character sum estimates obtained earlier in Part I [Acta Math. Hung. 118, No. 1-2, 129--148 (2008; Zbl 1164.11074)].
scientific article; zbMATH DE number 5355132