On disjoint sets

Abstract

Two sets of nonnegative integers $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are defined as \emph{disjoint}, if $\{A-A\}\bigcap\{B-B\}=\{0\}$, namely, the equation $a_i+b_t=a_j+b_k$ has only trivial solution. In 1984, Erd\H os and Freud [J. Number Theory 18 (1984), 99-109.] constructed disjoint sets $A,B$ with $A(x)>\varepsilon\sqrt{x}$ and $B(x)>\varepsilon\sqrt{x}$ for some $\varepsilon>0$, which answered a problem posed by Erd\H os and Graham. In this paper, following Erd\H{o}s and Freud's work, we explore further properties for disjoint sets. As a main result, we prove that, for disjoint sets $A$ and $B$, assume that $\{x_1<x_2<\cdots\}$ is a set of positive integers such that $\frac{A(x_n)B(x_n)}{x_n}\rightarrow 2$ as $x_n\to \infty$, then, (i) for any $0<c_1<c_2<1,$ $c_1x_n\le y\le c_2x_n$, we have $\frac{A(y)B(y)}{y}\rightarrow1$ as $n\rightarrow \infty$; (ii) for any $1<c_3<c_4<2,$ $c_3x_n\le y\le c_4x_n$, we have $A(y)B(y)=(2+o(1))x_n$ as $n\rightarrow \infty$.