Exact period recovery from a Fourier sample
= Exact period recovery from a Fourier sample
For the exact sample $c=sN/r$, reducing
$$
\frac cN=\frac sr
$$
recovers denominator $r/\gcd(s,r)$. Thus one sample reveals the full period exactly when $s$ is coprime to $r$; otherwise it reveals only a proper divisor and the algorithm must obtain more information.