ON ADDITIVE REPRESENTATION FUNCTIONS

For any finite abelian group$G$with$|G|=m$,$A\subseteq G$and$g\in G$, let$R_{A}(g)$be the number of solutions of the equation$g=a+b$,$a,b\in A$. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016,arXiv:1612.08722v1] proved that, if$m\geq 36$and$R_{A}(n)\geq 1$for all$n\in \mathbb{Z}_{m}$, then there exists$n\in \mathbb{Z}_{m}$such that$R_{A}(n)\geq 6$. In this paper, for any finite abelian group$G$with$|G|=m$and$A\subseteq G$, we prove that (a) if the number of$g\in G$with$R_{A}(g)=0$does not exceed$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$, then there exists$g\in G$such that$R_{A}(g)\geq 6$; (b) if$1\leq R_{A}(g)\leq 6$for all$g\in G$, then the number of$g\in G$with$R_{A}(g)=6$is more than$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$.

On the maximum values of the additive representation functions

Let [Formula: see text] and [Formula: see text] be infinite sequences of nonnegative integers. For a positive integer [Formula: see text] let [Formula: see text] denote the number of representations of [Formula: see text] as the sum of two terms from [Formula: see text]. Let [Formula: see text] denote the maximum value of [Formula: see text] up to [Formula: see text] and [Formula: see text] denote the distance of the sequences [Formula: see text] and [Formula: see text]. In this paper, we study the connection between [Formula: see text], [Formula: see text] and [Formula: see text]. We improve a result of Haddad and Helou about the Erdős–Turán conjecture.