New-Type Quasirandom Groups and Applications

This paper aims to introduce a more general definition of quasirandom groups and generalize several well-known results in the literature in this new setting. More precisely, let G be a semi-direct product of groups and \(X\subseteq G\) $X\subseteq G$ , we provide conditions such that one can find tuples \((x_0, \ldots , x_k)\in X^{k+1}\) $(x_0,\dots ,x_k)\in X^{k+1}$ satisfying \(x_1x_2\ldots x_k=x_0\) $x_1{x}_2\dots x_k=x_0$ or conditions to guarantee that the product set XX grows exponentially. In a special case of the group of rigid-motions on the plane over an arbitrary finite field, our results offer a reasonably complete description of structures of this group.