On a problem of Nathanson related to minimal asymptotic bases of order $h$

TL;DR

This paper characterizes partitions of natural numbers into subsets such that the union of their associated exponential sum sets forms a minimal asymptotic basis of a given order, generalizing previous results in the field.

Contribution

It provides new characterizations of partitions leading to minimal asymptotic bases, extending prior work by Chen, Ling, Tang, and Sun.

Findings

Characterizations of partitions producing minimal asymptotic bases

Generalization of previous results on asymptotic bases

Conditions for the union of exponential sum sets to be minimal

Abstract

For integer $h\geq2$ and $A\subseteq\mathbb{N}$, we define $hA$ to be all integers which can be written as a sum of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $n\in hA$ for all sufficiently large integers $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. For $W\subseteq\mathbb{N}$, denote by $\mathcal{F}^*(W)$ the set of all finite, nonempty subsets of $W$. Let $A(W)$ be the set of all numbers of the form $\sum_{f \in F} 2^f$, where $F \in \mathcal{F}^*(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup\cdots \cup W_h$ with the property that $A=A(W_1)\cup\cdots \cup A(W_{h})$ is a minimal asymptotic basis of order $h$. This generalizes a result of Chen and Chen, recent result of Ling and Tang, and also recent result of Sun.