Defining $\mathbb Z$ using unit groups

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\mathbb Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $\mathbb Z$. Namely, we prove that for a large collection of algebraic extensions $K/\mathbb Q$, $$ \{x \in \mathcal O_K \mid \forall \varepsilon \in \mathcal O_K^\times \ \exists \delta \in \mathcal O_K^\times : \delta -1 \equiv (\varepsilon -1)x\ (\mathrm{mod}\ (\varepsilon -1)^2)\} = \mathbb Z, $$ where $\mathcal O_K$ denotes the ring of integers of $K$. One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948.