论文标题
薄套件的添加剂完成
Additive completition of thin sets
论文作者
论文摘要
如果$ a+b $包含所有足够大的整数和$ a（x）b（x）/x \ rightarrow1 $，则两组$ a，b $的正整数称为\ emph {确切的添加剂}。令$ a = \ {a_1 <a_2 <\ cdots \} $为一组正整数。用$ a $ a $和$ a^*（x）$的计数函数表示$ a（x）$，用$ a \ bigcap [1，x] $中的最大元素。遵循Ruzsa和Chen-fang的工作，我们证明，对于确切的添加剂，$ a，b $与$ \ frac {a_ {n+1}}} {na_n} \ rightarrow \ rightarrow \ infty $，我们有$ a（x）b（x）b（x）-x \ ge ge ge ge \ frac {a^*（x）} {a（x）} +o \ left（\ frac {a^*（x）} {a（x）^2} \ right）$ as $ x \ rightarrow +\ rightarrow +\ infty $。另一方面，我们还构建了精确的添加剂$ a，b $与$ \ frac {a_ {n+1}}} {na_n} \ rightarrow \ infty \ infty $，以至于$ a（x）b（x）b（x）-x \ le le \ frac {a^*（x）} {a（x）}+（1+o（1））\ left（\ frac {a^*（x）} {a（x）^2} \ right）$保留了无限的许多正整数$ x $。
Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the counting function of $A$ and $a^*(x)$ by the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa and Chen-Fang, we prove that, for exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$, we have $A(x)B(x)-x\ge \frac{a^*(x)}{A(x)}+o\left(\frac{a^*(x)}{A(x)^2}\right)$ as $x\rightarrow +\infty$. On the other hand, we also construct exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$ such that $A(x)B(x)-x\le \frac{a^*(x)}{A(x)}+(1+o(1))\left(\frac{a^*(x)}{A(x)^2}\right)$ holds for infinitely many positive integers $x$.
