Exact additive complements

TL;DR

This paper investigates the structure of additive complements of positive integer sets, improving bounds on their sumset behavior and demonstrating near-optimality through examples.

Contribution

It advances the understanding of additive complements by refining bounds on the sumset and providing nearly optimal results with explicit examples.

Findings

Improved bounds on the sumset of additive complements.

Demonstrated near-optimality of the new bounds.

Extended previous results by Sárközy and Szemerédi, Chen and Fang.

Abstract

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