Complex approximations using algebraic integers

The problem of approximating complex numbers by elements of
Z[\omega]
, the algebraic integers of
Q(\omega)
, where
\omega
is a primitive
n
th root of unity, is considered. The motivating application is to reduce the dynamic range requirements of residue number system implementations of the discrete Fourier transform. Smallest error tolerances for the case of eighth roots of unity are derived using a geometric argument. Scale factors involved are reduced from
\alpha
to
\sqrt{\alpha}
for this case with roughly the same percentage errors. The case of sixteenth roots of unity gives even better range reductions and is considered only briefly.