One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies when $f$ lives on the integers, or on Euclidean space, but it is false for, say, the p-adic rationals, on which the characteristic function of the p-adic integers provides a counterexample.

One normally proves this phenomenon on Euclidean space through complex analysis, although it does follow from the real-variable argument on this post of Tao on Hardy's Uncertainty Principle.

For $f$ on the integers, it's even easier because only the zero polynomial has infinitely many roots (and this appears to me to be more or less the kind of input which goes into the other proofs as well).

What I want to know is whether there is a proof of this phenomenon -- namely, the refusal of a compactly supported functions F.T. to vanish on an open set -- which directly relates to the connectedness of the frequency space, if that is what is behind it.

Here is a failed effort in this direction...

You know that $f = f \chi$ for any function $\chi$ which is $1$ on the support of $f$. Then $\hat{f}$ should remain unchanged when convolved with the Fourier transform of $\chi$, but since $\hat{f}$ lives on the reals you would like to think that this convolution would partially fill up some open set connected to the boundary of the original support of $\hat{f}$ and thus enlarge the support of $\hat{f}$. This argument goes through as long as you don't get an absurd cancellation; but while $\hat{f}$ may be assumed positive, one has less freedom to renormalize $\chi$.

Does anyone know if there is a version of this argument which actually goes through? As stated it's no different than saying "$\hat{f} \ast \hat{\chi} = 0$ whenever $\chi$ lives away from the support of $f$; how weird..." It's possible my intuition here is all wrong and it's really the regularity of $\hat{f}$ which is completely behind the phenomenon, but I am at a loss for other examples of connected, locally compact abelian groups (are there any?) and I don't know how connectedness behaves with respect to Pontrjagin duality.