Defining the integers in large rings of a number field using one universal quantifier

Julia Robinson has given a first-order definition of the rational integers $\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall\exists\forall\exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8 variables, and where $F$ is a polynomial.
We show that for a large class of number fields, not including $\mathbb Q$, for every $\varepsilon>0$, there exists a set of primes $\mathcal S$ of natural density exceeding $1-\varepsilon$, such that $\mathbb Z$ can be defined as a subset of the “large” subring
$$ \{x\in K\colon\operatorname{ord}_\mathfrak px\geq0,\ \forall\,\mathfrak p\not\in\mathcal S\} $$
of $K$ by a formula where there is only one $\forall$-quantifier. In the case of $\mathbb Q$, we will need two quantifiers. We also show that in some cases one can define a subfield of a number field using just one universal quantifier. Bibl. – 18 titles.