On a problem of Chen and Fang related to infinite additive complements

On additive complement of a finite set

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

Exact additive complements

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. Sárközy and Szemerédi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved Sárközy and Szemerédi's bound. We further improve their estimate and show by an example that our result is nearly best possible.

A Note on Additive Complements

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results of Liu and Fang in 2016 and obtain some results on additive complements. For example, we prove that there exist additive complements $A$ and $B$ such that $\limsup_{x\to+\infty} A(x)B(x)/x= 2$ and $A(x)B(x) - x = 1$ for infinitely positive integers $x$.

关于平方数加性补集的一个注记

A note on additive complements of the squares

本文改进了平方数加性补集表示函数求和的下界，并以此在 Ben Green 提出的关于补集序列渐近性质的问题上取得进展。

Additive completition of thin sets

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 co…

关于非常一般除子对之补集的代数Green-Griffiths-Lang猜想

The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors

本文证明了 $\mathbb{P}^n$ 中总次数为 $2n$ 的非常一般除子对之补集在模去真闭子簇后代数双曲，从而为 Lang-Vojta 与 Green-Griffiths 猜想提供新证据，并部分推广了已有结果。