Can phase significantly concentrate a function's spectrum? @RajeshD - My motivation is to find better uncertainty principles, which definitely find applications in signal processing (analyzing sparse signals, compressed sensing, etc).

Can phase significantly concentrate a function's spectrum? Thanks! I'm not sure I agree with your gut, though. Taking $x_a(t):=\operatorname{sin}(2\pi at/n)$, simulations suggest that $\inf_{a,n}\|F_nx_a\|_1/\|F_n|x_a|\|_1=\pi/4$, where $F_n$ denotes the DFT over $\mathbb{Z}/n\mathbb{Z}$.