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arXiv · 2603.03472

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

Daniel Larsen
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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.

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Daniel Larsen. 2026-03-03. Three Questions of Erd\H{o}s-Nathanson on Asymptotic Bases of Order 2. https://arxiv.org/abs/2603.03472

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