Interpreting the projective hierarchy in expansions of the real line

TL;DR

This paper provides criteria for when expansions of the real line define complex algebraic structures, showing that certain expansions can define multiplication on real numbers, especially involving non-quadratic irrationals.

Contribution

It introduces a new criterion for expansions of the real line to define the image of the real field expanded by natural numbers under semialgebraic injections, highlighting the role of non-quadratic irrationals.

Findings

Expansions of the real line can define multiplication under specific conditions.

Non-quadratic irrational numbers play a key role in defining algebraic operations.

The paper characterizes when certain expansions lead to complex definability.

Abstract

We give a criterion when an expansion of the ordered set of real numbers defines the image of the expansion of the real field by the set of natural numbers under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number a, the expansion of the ordered Q(a)-vector space of real numbers by the set of natural numbers defines multiplication on the real numbers.