Fourier analysis, orthogonality, and Plancherel for finite abelian groups

Question

I am reading an outstanding paper by Bateman and Katz, improving the best known bounds on the cap set problem (Roth's theorem over $\mathbb{F}_3^N$).

The paper contains some technical lemmas for which I believe there must be an excellent geometric intuition -- which I am afraid I am missing.

Excerpting from, and simplifying, Section 8 of their paper, let $A \subset Y := \mathbb{F}_3^n$

be some subset, and also write $A(x)$ for the characteristic function of $A$. Let $H$ be a subspace of $Y$ and let $H^{\perp}$ be its annihilator. For $h \in H$, write $A_{H, h} := A \cap (H^{\perp} + h)$. Then we have a version of Plancherel

$$\sum_{h \in H} |\widehat{A}(h)|^2 = |H| 3^{-2N} \sum_{h \in H} |A_{H, h}|^2,$$

and further, if $K$ is a subspace of $Y$ containing $H$,

$$\sum_{0 \neq k \in K} |\widehat{A}(k)|^2 = \sum_{0 \neq h \in H} |\widehat{A}(h)|^2 + \frac{1}{|H|} \sum_{h \in H} \sum_{0 \neq k \in K/H} |\widehat{A}_{H, h}(k)|^2.$$

There are other interesting related formulas as well. The authors remark that the latter equality "can simply be thought of as Plancherel for a 'local Fourier transform' of $A$. Here, we localize to the translates of $H^{\perp}$."

I can verify the identities readily enough, but I feel like there should be some excellent geometric intuition to be had here, with which all of these equalities are obvious. Is there anything that can be said which will render these equalities transparent? Perhaps some elaboration of the authors' remark?

Thank you!

Answer 1

I don't know whether it would be perceived as "geometric", but an intuition that "works for me" on this and related matters is that "Fourier analysis" on finite abelian groups is "abelian" Fourier analysis (e.g., on products of circles or lines, in the classical analytic scenarios) without the need to "do analysis".

Thus, the once-scandalous fact that the Fourier transform of Dirac delta is the identically-one function is true, without scandal.

More generally/similarly, the Fourier transform of the "delta" of a subgroup H is the collection of characters of the ambient G which restrict to the trivial character on H.

Translations twist by character-values...

Fourier transform of characteristic function of a set $A$ is the sum of the Fourier transforms of the deltas of points in the set. If the set contains or is contained in translates of subgroups, various reasonable identities hold.

Possibly this is all-to-obvious... and doesn't count as "geometric"... but this style of explanation works(-for-me) very well.