Three Questions of Erd\H{o}s-Nathanson on Asymptotic Bases of Order 2

TL;DR

This paper investigates the robustness properties of asymptotic bases of order 2, demonstrating independence of key properties under weaker growth conditions and providing new constructions via an inductive scheme.

Contribution

It proves that certain properties of asymptotic bases are independent under weaker growth rates, extending prior results by Erdős and Nathanson.

Findings

Properties are independent for weaker growth rates

Constructs bases with specific properties via inductive scheme

Shows divergence in behavior based on growth rate

Abstract

We study three natural properties that measure the robustness of asymptotic bases of order 2: having divergent representation function, being decomposable as a union of two bases, and containing a minimal basis. Erd\H{o}s and Nathanson showed that sufficiently rapid growth of the representation function (specifically, $r_A(n) \ge C \log n$ for appropriate $C$) implies both decomposability and the existence of a minimal basis. We prove that for weaker growth rates, these three properties are independent. The construction proceeds via an inductive scheme on exponentially growing intervals.