Additive completition of thin sets

TL;DR

This paper investigates the structure of exact additive complements of positive integers, establishing bounds on their counting functions and constructing examples with specific asymptotic behaviors.

Contribution

It provides new bounds on the sum of counting functions for exact additive complements and constructs examples demonstrating these bounds are tight.

Findings

Established lower bounds for $A(x)B(x)-x$ in terms of $a^*(x)$ and $A(x)$.

Constructed examples of exact additive complements with prescribed asymptotic properties.

Extended previous work by Ruzsa and Chen-Fang on additive complements.

Abstract

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 $x$.