Quantum Fourier transform of a periodic coset state
= Quantum Fourier transform of a periodic coset state
Let $r$ divide $N$ and
$$
|\alpha\rangle=\frac1{\sqrt{N/r}}
\sum_{j=0}^{N/r-1}|x_0+jr\rangle.
$$
The geometric sum in the quantum Fourier transform vanishes unless $c$ is a multiple of $N/r$. Thus measurement after the transform returns the $r$ values
$$
c=0,\frac Nr,\ldots,(r-1)\frac Nr
$$
with equal probabilities $1/r$.