I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the power $\alpha$, it is still as close as $\delta$ to the actual value of $\omega^\alpha$.
If we have an $n$th root of unity, as:
$$\omega = e^{2 \pi i / n}$$
...and we want an approximation of $\omega$, as:
$$\omega_{\text{approx}} = \omega + \epsilon$$
...such that
$$\left| \Re \left( \left(\omega_{\text{approx}} \right)^\alpha - \omega^\alpha \right) \right| < \delta$$
...and
$$\left| \Im \left( \left(\omega_{\text{approx}} \right)^\alpha - \omega^\alpha \right) \right| < \delta$$
Can we get $\epsilon$ in terms of $n$, $\omega$, $\alpha$, and $\delta$?
Given $\alpha$ a positive integer and $0<\delta\le1$, set $$ \epsilon=(1+\delta)^{1/\alpha}-1. $$ We claim that if $|z-\omega|<\epsilon$, then $|z^\alpha-\omega^\alpha| < \delta$; in particular, the real and imaginary parts of that latter difference are both less than $\delta$.
To see this, consider the function $$ f(z) = z^{\alpha-1} + z^{\alpha-2}\omega + \cdots + z\omega^{\alpha-2} + \omega^{\alpha-1}, $$ so that $$ (z-\omega)f(z) = z^\alpha-\omega^\alpha. $$ Therefore $$ |z^\alpha-\omega^\alpha| = |z-\omega||f(z)| < \epsilon(|z|^{\alpha-1} + |z|^{\alpha-2}|\omega| + \cdots + |z||\omega|^{\alpha-2} + |\omega|^{\alpha-1}). $$ Note that $|\omega|=1$, while $|z|<1+\epsilon$ since $|z-\omega|<\epsilon$. Therefore $$ |z^\alpha-\omega^\alpha| < \epsilon((1+\epsilon)^{\alpha-1} + (1+\epsilon)^{\alpha-2} + \cdots + (1+\epsilon) + 1) = (1+\epsilon)^\alpha-1 = \delta, $$ as desired.