论文标题
在脱节集上
On disjoint sets
论文作者
论文摘要
两组非负整数$ a = \ {a_1 <a_2 <\ cdots \} $和$ b = \ {b_1 <b_1 <b_2 <\ cdots \} $被定义为\ emph {disehoint}也就是说，方程$ a_i+b_t = a_j+b_k $只有微不足道的解决方案。 1984年，Erd \ H OS和Freud [J.数字理论18（1984），99-109。]构建的不相交集合$ a，b $，$ a（x）> \ varepsilon \ sqrt {x} $和$ b（x）> \ varepsilon \ sqrt \ sqrt {x} $，对于某些$ \ varepsilon> 0 $，这是一个问题，这是一个问题，并被一个问题。在本文中，遵循ErdőS和弗洛伊德的工作，我们探讨了脱节集的更多属性。作为主要结果，我们证明，对于脱节设置$ a $ a和$ b $，假设$ \ {x_1 <x_1 <x_2 <\ cdots \} $是一组积极的固定器，以至于$ \ frac {a（x_n）b（x_n）b（x_n）} $ 0 <c_1 <c_2 <1，$ $ $ c_1x_n \ le y \ le y \ le c_2x_n $，我们有$ \ frac {a（y）b（y）b（y）} {y} {y} \ rightarrow1 $ as $ n \ rightArrow \ rightArrow \ rightArrow \ infty \ infty $; （ii）对于任何$ 1 <c_3 <c_4 <2，$ $ C_3X_N \ le y \ le y \ le c_4x_n $，我们有$ a（y）b（y）b（y）=（2+o（1））x_n $ as $ n \ rightArrow \ rightArrow \ rightarrow \ infty $。
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 Erdős 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 $n\rightarrow \infty$.
