\textit{T. Tao}'s uncertainty principle [Math. Res. Lett. 12, No. 1, 121--127 (2005; Zbl 1080.42002)] states that if \(p\) is prime then, for the cyclic group \(\mathbb{Z}/p\) of integers modulo \(p\), the support of any function \(f\) on \(\mathbb{Z}/p\) and its group Fourier transform \(\widehat{f}\) must satisfy \[ |\text{supp}\, f|+|\text{supp}\, \widehat{f}|\geq p+1 \] where \(|\cdot|\) denotes the number of elements of a subset. The result boils down to nonvanishing of a certain Vandermonde determinant. In the present work this result is extended to the cyclic group \(\mathbb{Z}/m\) in which \(m\) is composite and the support of \(f\) has a certain structure property related to vanishing exponential sums. To describe this property, recall that a dominant weight is a multi-index \(\kappa=(\kappa_1,\dots,\kappa_n)\) such that \(\kappa_i\geq \kappa_{i+1}\). By the Weyl character formula, each dominant weight \(\lambda\) corresponds to a unique irreducible character \(\chi_\lambda\) of the group \(U(n)\) of unitary \(n\times n\) matrices and \(\chi_\lambda\) is a function on the \(n\)-torus such that \(\chi_\lambda(1)={\prod_{1\leq i