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Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2

Daniel Larsen

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. Erdős 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.

Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2

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. Erdős and Nathanson showed that sufficiently rapid growth of the representation function (specifically, for appropriate ) 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.
Paper Structure (8 sections, 7 theorems, 23 equations)

This paper contains 8 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Decomposable Case
  3. Bases with no minimal subbasis
  4. Bases with a minimal subbasis
  5. Indecomposable case
  6. Bases with no minimal subbasis
  7. Bases with a minimal subbasis
  8. Construction

Key Result

Theorem 1

There exist asymptotic bases satisfying all possible combinations of (P1), (P2), and (P3).

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 3 more