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.
Categories: math.NT