$L^p$ averages of the Fourier transform in finite fields

The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. In general, sets with good uniform bounds for the Fourier transform are less structured, more `random', and can often be analysed more easily. In many cases obtaining good \emph{uniform} bounds is not possible, even if `most' points admit good pointwise bounds. Motivated by this, we propose a more nuanced approach where one seeks to bound the $L^p$ averages of the Fourier transform instead of only the maximum. We explore this idea by considering several examples and find that a rich theory emerges. Further, we provide various applications of this new approach; including to sumset type problems, the finite fields distance conjecture, and the problem of counting $k$-simplices inside a given set.
Comment: 31 pages, 1 figure. Minor changes and references added