A Dichotomy for $k$-automatic expansions of Presburger Arithmetic

TL;DR

This paper establishes a dichotomy for $k$-automatic subsets of natural numbers, showing they are either all definable in an expanded Presburger arithmetic or equivalent to a structure with exponential growth.

Contribution

It proves a new dichotomy characterizing the definability of $k$-automatic sets within expanded Presburger arithmetic frameworks.

Findings

Either all $k$-automatic subsets are definable in the expanded structure with $X$

Or $( N, +, X)$ has the same definable sets as $( N, +, k^{ N})$

The dichotomy clarifies the logical complexity of $k$-automatic sets

Abstract

Let $k\ge 2$ and let $X$ be a subset of the natural numbers that is $k$-automatic and not eventually periodic. We show that the following dichotomy holds: either all $k$-automatic subsets are definable in the expansion of Presburger arithmetic in which we adjoin the predicate $X$, or $(\mathbb{N},+,X)$ has the same definable sets as $(\mathbb{N},+,k^{\mathbb{N}})$.