The New Upper Bounds of Some Ruzsa Numbers $R_m$

Abstract:

      For $A\subseteq {\mathbf{Z}}_m$ and $n\in {\mathbf{Z}}_m$, let $\sigma_A(n)$ be the number of solutions of equation $n=x y, x,y\in A$. Given a positive integer $m$, let $R_m$ be the least positive integer $r$ such that there exists a set $A\subseteq {\mathbf{Z}}_m$ with $A A={\mathbf{Z}}_m$ and $\sigma_A(n)\leq r$. Recently, Chen Yonggao proved that all $R_m\leq 288$. In this paper, we obtain new upper bounds of some special type $R_{kp^2}$.