Robust additive bases without minimal subbases

TL;DR

The work resolves a question of Erdős and Nathanson by constructing an additive basis of order two whose representations grow at least like a constant times $\log m$ for large $m$, yet which contains no minimal subbasis. The authors implement a block-wise, probabilistic construction across exponentially growing intervals, thinning and then thickening each block to control $r_A(m)$ while preserving fragile, non-minimal structure in potential subbases. Central to the argument are concentration bounds (Chernoff) and almost-sure (Borel–Cantelli) controls, plus a dihedral-group counting framework to bound higher-order representations and ensure independence of blocks. The resulting set $A$ demonstrates that even logarithmic representation growth does not guarantee the existence of a minimal subbasis, advancing the understanding of additive-basis structure and minimality phenomena.

Abstract

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$ minimal for the property that $D+D$ contains all sufficiently large positive integers.

Theorems & Definitions (18)

• Theorem 1
• Lemma 2
• proof
• Corollary 3
• Lemma 4
• proof
• Proposition 5
• proof
• Lemma 6
• proof