In reading the literature one encounters countless examples of Voronoi formulas, i.e., formulas that take a sum over Fourier coefficients, twisted by some character, and controlled by some suitable test function, and spits out a different sum over the same Fourier coefficients, twisted by some different characters, and this time controlled by some integral transform of the test function.

The reason one wants to do this in practice is that the second sum is somehow better, of course, which in my (admittedly limited) experience tends to boil down to the length of the second sum having changed significantly to the better.

I'll give an example (from Xiaoqing Li's Bounds for GL(3)×GL(2) L-functions and GL(3) L-functions, because it is what I happen to have in front of me). In this case we have the GL(3) Voronoi formula $$ \sum_{n > 0} A(m, n) e\Bigl( \frac{n \bar d}{c} \Bigr) \psi(n) \sim \sum_{n_1 \mid c m} \sum_{n_2 > 0} \frac{A(n_2, n_1)}{n_1 n_2} S(m d, n_2; m c n_1^{-1}) \Psi \Bigl( \frac{n_2 n_1^2}{c^3 m} \Bigr), $$ where $\psi$ is some smooth, compactly supported test function, $\Psi$ as suggested above is some integral transform of it, $A(m, n)$ are Fourier coefficients of (in this case) an SL(3) Maass form, $(d, c) = 1$, and $d \bar d \equiv 1 \pmod{c}$.

(I've omitted lots of details here, but the details, I think, aren't relevant to my question.)

Doing so essentially transforms the $n$-sum into the $n_2$-sum, where, as is evident in the formula, the $n_2$-sum has a very different argument in its test function.

What happens in practice now is that, once we get to a point where applying the Voronoi formula is appropriate, we transform the sum and study the integral transform, chiefly by means of stationary phase analysis in order to find what length of the new $n_2$-sum is.

In the particular example at hand, this, after identifying the stationary phase and playing along, this takes us from an $n$-sum on $N \leq m^2 n \leq 2 N$, i.e., $n \sim \frac{N}{m^2}$, to an $n_2$-sum on $$ \frac{2}{3} \frac{N^{1/2}}{n_1^2} \leq n_2 \leq 2 \frac{N^{1/2}}{n_1^2}, $$ i.e., $n_2 \sim \frac{N^{1/2}}{n_1^2}$, which then means that the arguments in the test function are now of size $\frac{N^{1/2}}{c^3 m}$.

I can go through the motions of performing this stationary phase analysis and so on, but my question is this: **is there any intuition to be had about how and in what way these Voronoi formulas alter the lengths of sums?**