Arithmetic unlikely intersections in powers of the multiplicative group

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field.

Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its ``size" should not be too big compared to the ``complexity" of the subgroup scheme.

We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-Mérai-Ostafe-Sha.

We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.