TITLE: Question concerning Tao's proof of uncertainty principle in Z/pZ
QUESTION [5 upvotes]: In his paper "An Uncertainty principle for cyclic groups of prime order", in the proof of Theorem 1.1, while proving the converse statement, Tao makes the following statement- "It will suffice to prove the claim when $|A| + |B| = p+1$, since the claim for $|A|+|B| > p+1$ then follows by applying the claim to subsets $A',B'$ of $A,B$ respectively for which $|A'|+|B'| = p+1$, and then taking generic linear combinations as $A',B'$ vary."
This part is not very clear to me and I was trying to write down the details properly for this, and would really appreciate some help.
Claim: If $A,B$ are nonempty subsets of $\mathbb{Z}/p\mathbb{Z}$ such that $|A|+|B|\geq p+1$ then we can find function $f$ such that $\mathrm{supp} f= A, \mathrm{supp} \hat{f} = B$.
To this claim, Tao makes the aforementioned remark, and my attempt is to prove the assertion that it suffices to prove this claim for the case $|A|+|B|=p+1$. I am trying to prove the claim when $|A|+|B| > p+1$ given that the assertion is true for subsets $A',B'$ of $A,B$ respectively such that $|A'|+|B'|=p+1$.
My attempt: Take all possible combinations of subsets $\{\{A_{i},B_{i}\}| i \in I\}$ such that $|A_{i}|+|B_{i}|=p+1$, $A_{i},B_{i}$ are subsets of A and B respectively, for each $\{A_{i},B_{i}\}$ we would get $f_{i},\hat{f_{i}}$ such that $\mathrm{supp} f_{i}=A_{i}, \mathrm{supp} \hat{f_{i}}=B_{i}$. Now, define a new function $f$ as follows
$$f := c_{1}f_{1} + c_{2}f_{2} + \cdots + c_{r}f_{r}$$ where $r=|I|$. Then it is obvious that $\mathrm{supp} f \subseteq A$, I want to show the other way around that $A \subseteq \mathrm{supp}f$.
I am not sure how to conclude my claim ($\mathrm{supp}f=A$), and is this what Tao had in mind or am I missing some much simpler.

REPLY [2 votes]: For each $x \in A$, the set
$$H_x=\{(c_1,\ldots,c_r) \in {\mathbb R}^r \,: \, \sum_{i=1}^r c_i f_i(x)=0\}$$
is a hyperplane (of volume zero) in ${\mathbb R}^r$. The union
$\cup_{x \in A} H_x$ still has volume zero, and we just need to choose $(c_1,\ldots,c_r)$ outside this union.
To use another interpretation of the term "generic", observe that each $H_x$ is a closed nowhere dense set in ${\mathbb R}^r$, so the union
$\cup_{x \in A} H_x$ is also a closed nowhere dense set.