I am curious about a set of related elementary questions in algebraic number theory about translations of rings of integers of number fields by elements algebraic over them. The most basic one is this: suppose that $L / K$ is an extension of number fields, and $\alpha$ is an element of the ring of integers $\mathcal{O}_{L}$ with $\alpha \notin K$. Then for how many elements $t \in \mathcal{O}_{K}$ is $\alpha + t$ a unit in $\mathcal{O}_{L}$? I strongly suspect that there can only be finitely many such $t \in \mathcal{O}_{K}$ (it is easy to show that this is the case when $K = \mathbb{Q}$ or $K$ is an imaginary quadratic field so that it has finitely many units), but I don't see how to prove this.

A more general question would be the following. Given a finite set $S$ of primes of $K$ and $\alpha \in \mathcal{O}_{L} \setminus \mathcal{O}_{K}$ as before, for how many $t \in \mathcal{O}_{K}$ is the ideal $(\alpha + t) \subset \mathcal{O}_{L}$ divisible only by primes lying over $S$?