This is a followup to this earlier question
Let $f:\mathbb{Z}\rightarrow \{\pm 1\}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it helps. To avoid divisibility issues, assume that $N$ is is not divisible by any integer in $[1,m]$. One could, for example, take $N=m\#-1,$ where $m\#$ is the $m^{th}$ primorial.
Define the fourier transform $\widehat{f}$ on $[0,1)$ by $\widehat{f(t)}=\sum_{n\in \mathbb{Z}} f(n)~e^{-2i \pi n t}.$
Now let $v$ be a positive integer $\geq 2,$ and let the "projected" function be $$ f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right. $$ Of course $f_1$ is simply $f.$ Can one obtain a nontrivial bound of the form $$ \sum_{v=1}^m \mid \sum_{n \in \mathbb{Z}} f_v(n) \mid^2 {\geq} A(N,m) \int_0^1 \mid\widehat{f(t')} \mid^2 \,dt' $$ or similar, using some kind of uncertainty relation.
We can take $m\ll N,$ a fractional power of $N$ or even a power of $\log N.$
