L^p averages of the Fourier transform in finite fields

2024/07/11 by Fraser, Jonathan M. · 3 citations
#11L40 #43A25 #43A46 #52C35 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #primary: 52C10 #secondary: 11B30

paper · doi:10.48550/arxiv.2407.08589

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^∞ 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 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.

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