Sparsity of DFT of powers of random sequence

Question

Given a randomly chosen finite sequence of complex values ($a_0,a_1 \dots a_{N-1}$) such that their DFT is sparse, can anything be said about the sparsity of the DFT of the sequences ($a_0^{\frac{k}{N}}, a_1^{\frac{k}{N}} \dots a_{N-1}^{\frac{k}{N}}$) for $k : (1,2 \dots N-1$)? Is my intuition right in saying that the sequences would have a high probability of having sparse DFT?