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

TL;DR

This paper extends Chebotarev's theorem to certain matrices associated with composite numbers of the form pq, and applies this to establish an uncertainty principle for cyclic groups of such orders.

Contribution

It generalizes Chebotarev's theorem to matrices of order pq with specific prime conditions and derives an uncertainty principle for cyclic groups of these orders.

Findings

Principal submatrices are non-singular for n=pr under certain conditions

Established an uncertainty principle for cyclic groups of order pq

Extended Chebotarev's theorem to composite orders with prime factors

Abstract

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 $\mathbf{Z}_r^*$. As an application, an uncertainty principle for cyclic groups of order $n$ is established when $n=pr$ as described above.