Lᵖ 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 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 Lᵖ 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.