Let $S\subset \overline{\mathbf{Q}}\subset \mathbf{C}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers.

Let $U$ be a non-empty open subset in the Euclidean topology on $\mathbf{C}$.

Does $U$ contain infinitely many solutions to the unit equation. That is, does the intersection $S\cap U$ contain infinitely many elements?

I also posted this question on stackexchange yesterday, but didn't get an answer.