Robust additive bases without minimal subbases

TL;DR

This paper constructs a set of positive integers with a logarithmic lower bound on representations but no minimal subset whose sumset covers all large integers, challenging assumptions about additive bases.

Contribution

It introduces a set with complex additive properties, showing the non-existence of minimal subbases even when representation counts grow logarithmically.

Findings

Existence of a set with representation count ≥ log m

No minimal subset D with D+D covering all large integers

Challenges previous assumptions about additive bases

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.