On a problem of Nathanson

TL;DR

This paper addresses a problem posed by Nathanson regarding the characterization and properties of minimal asymptotic bases of a given order, which are sets of nonnegative integers with specific additive covering properties.

Contribution

The paper provides a resolution to Nathanson's problem by establishing new results on the structure and existence of minimal asymptotic bases of order h.

Findings

Resolved Nathanson's problem on minimal asymptotic bases

Characterized the structure of minimal asymptotic bases

Proved existence results for minimal asymptotic bases

Abstract

A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. In this paper, we resolve a problem of Nathanson on minimal asymptotic bases of order $h$.