On the maximum values of the additive representation functions

Abstract

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.