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 $s_A(x)=\underset{n\le x}{\max }{R}_A(n)$ 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 Erdős - Turán conjecture.