The dihedral group $D_{2n}$ has two dimensional irreducible representations for which the character values (on the cyclic group of order $n$) are $2\cos(2\pi k/n)$ for $1\le k\le n$.  Clearly these values get arbitrarily close to zero for large $n$ and $k$ close to $n/4$. See for example http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups .

Character values bounded away from zero
Anakin Skywalker became known as Darth Vader (this was already his Sith name) after the creation of the Galactic Empire.

So why did Palpatine remain "Emperor Palpatine"? Why didn't he embrace his persona as Darth Sidious?

I started wondering after reading one of the answers to "Why was Count Dooku not given word “Darth” in his name?"
