A Second Note on a Result of Haddad and Helou

Abstract:

      Let $K$ be a finite field of characteristic $\neq 2$ and $G$ the additive group of $K\times K$. Let $k_{1}$, $k_{2}$ be integers not divisible by the characteristic $p$ of $K$ with $(k_{1}, k_{2})=1$. In 2004, Haddad and Helou constructed an additive basis $B$ of $G$ for which the number of representations of $g\in G$ as a sum $b_{1}+b_{2}(b_{1}, b_{2}\in B)$ is bounded by 18. For $g\in G$ and $B\subset G$, let $\sigma_{k_{1}, k_{2}}(B, g)$ be the number of solutions of $g=k_{1}b_{1}+k_{2}b_{2}$, where $b_{1}, b_{2}\in B$. In this paper, we show that there exists a set $B\subset G$ such that $k_{1}B+k_{2}B=G$ and $\sigma_{k_{1}, k_{2}}(B, g)\leqslant 16$.