Let $\zeta$ be a primitive $2^k$th root of unity, and consider the ring $ \mathbb{Z}[\zeta, \frac{1}{2}] \subset \mathbb{C}$. Are there any elements of absolute value 1 other than the powers of $\zeta$?
Note that if $\frac{1}{2}$ were replaced by $\frac{1}{5}$, then the number $\frac{4}{5} + \frac{3}{5} i$ would be an example.
I can prove that the answer is 'no' for $k \leq 3$. In particular, for $k = 3$ we want to find integers $a, b, c, d$ and positive integer $n$ such that:
$$ (a + b \zeta + c \zeta^2 + d \zeta^3)(a + b \zeta^{-1} + c \zeta^{-2} + d \zeta^{-3}) = 4^n $$
Expanding this yields:
$$ a^2 + b^2 + c^2 + d^2 + (ab + bc + cd - da) \sqrt{2} = 4^n $$
By Jacobi's four-square theorem, either all but one of $a,b,c,d$ is zero (and we get a cyclotomic unit) or $a^2 = b^2 = c^2 = d^2 = 4^{n-1}$. In this case, $|ab| = |bc| = |cd| = |da|$ and an even number of them are negative. It follows that the coefficient of $\sqrt{2}$ is nonzero, whence we derive a contradiction.
Does this result hold for all $k$?