Haddad和Helou定理的另一结论

A Second Note on a Result of Haddad and Helou

中文关键词:  加法基  表示函数

英文关键词:additive basis  representation function

中文摘要:

      令$K$是特征不为$2$有限域, $G$为$K\times K$. 令$k_{1}$, $k_{2}$是不被 $K$的特征 $p$整除的整数, 且 $(k_{1}, k_{2})=1$. 2004 年, Haddad 和Helou 构造了$G$ 的加法基$B$, 使得对任意$g\in G$, $g=b_{1}+b_{2}(b_{1}, b_{2}\in B)$解的个数小于18. 对任意$g\in G$, $B\subset G$, 令$\sigma_{k_{1}, k_{2}}(B, g)$是$g=k_{1}b_{1}+k_{2}b_{2}$ 解的个数, 其中$b_{1}, b_{2}\in B$. 本文, 我们证明存在集合$B\subset G$使得$k_{1}B+k_{2}B=G$, 且$\sigma_{k_{1}, k_{2}}(B, g)\leqslant 16$.

英文摘要:

      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 $\sigma_{k_{1}, k_{2}}(B, g)\leqslant 16$.