The additive group of the rationals does not have an automatic presentation

TL;DR

This paper proves that the additive group of the rationals cannot be described by an automatic presentation, using combinatorial methods and Freiman's theorem, and extends the result to other abelian groups with similar properties.

Contribution

It establishes the non-existence of automatic presentations for the additive group of rationals and related abelian groups, employing combinatorial techniques.

Findings

The additive group of rationals does not admit an automatic presentation.

The proof extends to certain torsion-free abelian groups with p-divisibility.

Uses Freiman's theorem on small doubling sets in the proof.

Abstract

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $p$-divisible for infinitely many primes $p$, or $Q/Z$. The proof is combinatorial and uses most notably Freiman's theorem on sets with small doubling.