Let say Z is a sum of n-roots of unity and thus an algebraic integer, and D is an rational integer. If |z+D| is an integer, what can we conclude regarding Z? can we say |Z| is integer? Another related question is, for which non-zero D, we can conclude |Z| is integer from the given that |Z+D| is an integer?
If $D$ is a nonzero rational number and $R$ is a positive number, the complex numbers $z$ with $|z+D|=R$ form the circle of radius $R$ centred at $-D$. The intersection of this with the circle $|z| = k$ (if nonempty) consists of one or two points satisfying the quadratic $D z^2 + (D^2 + k^2 - R^2) z + k^2 D = 0$. So if $z$ with $|z+D|^2$ rational has degree $> 2$ over the rationals, $|z|^2$ can't be a rational number.