When is an automatic set an additive basis?

TL;DR

This paper characterizes when k-automatic sets of natural numbers form additive bases, providing an effective characterization and an algorithm to determine the minimal order of such bases.

Contribution

It offers a complete characterization and an algorithmic method to identify when k-automatic sets are additive bases and to find their minimal order.

Findings

Characterization of k-automatic additive bases

Effective criteria for identifying additive bases

Algorithm to compute the minimal order of the basis

Abstract

We characterize those $k$-automatic sets $S$ of natural numbers that form an additive basis for the natural numbers, and we show that this characterization is effective. In addition, we give an algorithm to determine the smallest $j$ such that $S$ forms an additive basis of order $j$, if it exists.