Fourier analysis is used in additive combinatorics as a way to detect structure in sets(Roth’s theorem, Szemeredi’s Theorem, Erdos-Szemeredi conjecture, Green-Tao Theorem etc). In particular(from what I understand) Fourier analysis is used to detect quasirandomness in sets and thats what gets abused in the proofs of said theorems.
I’ll admit I haven’t thought too much about it, but what’s stopping us from using a wavelet designed to detect these kinds of structures and apply multiresolution analysis to the indicator function for some set A with suitable density? The way multiresolution analysis is set up is such that the mother wavelet is scaled and shifted across a given domain so it “detects” the wavelet structure at different resolutions, which would seem to lend itself to detecting structures like arithmetic progressions if the chosen wavelet is designed in a particular way. Maybe it would be better at detecting these structures than a Fourier analytic techniques would? This seems like the kind of thing that if it were possible would have already been investigated, so what am I missing?
Perhaps this is more a question about what makes the Fourier transform in particular good at detecting these structures and subsequently realizing the wavelet decompositions do not possess these qualities.