Definability of the natural numbers in totally real towers of nested square roots

Abstract:

For the ring of integers $\mathcal {O}$ of a totally real algebraic field, Julia Robinson defines a set $A(\mathcal {O})$ such that either $A(\mathcal {O})=\{+\infty \}$ or it is an interval in $\mathbb {R}$. She then proves that if this set has a minimum, then the natural numbers can be defined in $\mathcal {O}$, and hence $\mathcal {O}$ has undecidable first-order theory. All known examples are such that $A(\mathcal {O})$ has a minimum which is either $4$ or $+\infty$. In this work, we construct two infinite families of subrings of such rings for which $\inf (A(\mathcal {O}))$ is strictly between $4$ and $+\infty$. In one family, the infimum is a minimum, whereas in the other family it is not, but we can still define the natural numbers in this case.