Chebotarev's theorem for groups of order $pq$ and an uncertainty principle

Let $p$ be a prime number and ${\zeta }_p$ a primitive $p$-th root of unity. Chebotarev’s theorem states that every square submatrix of the $p\times p$ matrix $({\zeta }_p^{ij}{)}_{i,j=0}^{p-1}$ is non-singular. In this paper we prove the same for principal submatrices of $({\zeta }_n^{ij}{)}_{i,j=0}^{n-1}$, when $n=pr$ is the product of two distinct primes, and $p$ is a large enough prime that has order $r-1$ in ${\mathbf{Z}}_r^{\ast }$. As an application, an uncertainty principle for cyclic groups of order $n$ is established when $n=pr$ as described above.