Undefinability of multiplication in Presburger arithmetic with sets of powers

TL;DR

This paper proves that adding sets of powers to Presburger arithmetic cannot define multiplication and that any definable image of such sets has zero density, extending known dichotomies in o-minimal structures.

Contribution

It establishes the undefinability of multiplication in Presburger arithmetic expanded by sets of powers, using density and dichotomy arguments.

Findings

Any Presburger-definable image of sets of powers has zero natural density.

Expansion by sets of powers does not define multiplication.

A dichotomy similar to o-minimal structures applies to Presburger expansions.

Abstract

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.