I will copy-paste the statement and the theorem from this paper by Tao about an uncertainty principle for groups of prime order. http://arxiv.org/pdf/math/0308286.pdf
Theorem 1.1: Let $p$ be a prime number. If $f: \mathbb{Z}/p\mathbb{Z} \to \mathbb{C}$ is a non-zero function, then $$|\mathrm{supp}(f)|+|\mathrm{supp}(\tilde f)| \geq p+1$$ Conversely, if $A$ and $B$ are two non-empty subsets of $\mathbb{Z}/p\mathbb{Z}$ such that $|A|+|B| \geq p+1$, then there exists a function $f$ such that $\mathrm{supp}(f)=A$ and $\mathrm{supp}(\tilde f)=B$.
He goes on to say:
The informal explanation of this principle is that the class of functions $f$ from $\mathbb{Z}/p\mathbb{Z} \to \mathbb{C}$ has exactly $p$ degrees of freedom. Requiring that $\mathrm{supp}(f) = A$ takes away $p −|A|$ of these degrees, while requiring that $\mathrm{supp}(\tilde f) =B$ takes away another $p −|B|$. The uncertainty principle is then a statement that the Fourier basis (of exponentials) and the physical space basis (of Dirac deltas) are “totally skew” (or more precisely, that all the minors of the exponential basis matrix $(\exp({\frac{2 \pi ij k}{p}}))_{0\leq j,k \leq p}$ are non-zero).
What does he mean by $p$ degrees of freedom? Is he thinking of the set $\{f \mid f: \mathbb{Z}/p\mathbb{Z} \to \mathbb{C}\}$ as a $p$-dimensional vector space? Is that what he means? If yes, then why is the next claim true?
How his second claim about the uncertainty principle is demonstrated?