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

Abstract

The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. If a set admits optimal $L^\infty$ bounds on its Fourier transform (that is, it is a Salem set), then it can often be analysed more easily. However, 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 framework where one systematically studies the $L^p$ averages of the Fourier transform and keeps track of how good the $L^p$ bounds are as a function of $p$. This captures more nuanced information about a set than, for example, asking whether it is Salem or not. We explore this idea by considering several examples and find that a rich theory emerges. Further, we provide various applications of this approach; including to sumset type problems, the finite fields distance conjecture, and the problem of counting $k$-simplices inside a given set. Our typical application is of the form:~if a set admits good $L^p$ bounds on its Fourier transform, then we are able to make strong geometric conclusions.