On the maximum values of the additive representation functions

Abstract

Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and $\displaystyle d_{A,B}(x) = \max_{\hbox{t: $a_{t} \le x$ or $b_{t} \le x$}}|a_{t} - b_{t}|$. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. We improve a result of Haddad and Helou about the Erd\H{o}s - Tur\'an conjecture.