# Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ 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?

Since there werent't any replies, I also asked this question on Mathoverflow.

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mathoverflow.net/questions/78876/… to be precise, where there is now an answer posted. –  Gerry Myerson Oct 23 '11 at 11:49