Abstract
In the previous two chapters, we have gained a significant amount of understanding about the L p -average decay for the Fourier transform of characteristic functions of convex sets and considered some applications to problems in lattice point counting and discrepancy theory. In this chapter we consider more elaborate applications of average decay in number theory where the discrepancy function needs to be estimated for almost every rotation instead of averaging over rotations in some L p -norm. This naturally leads us to the examination of certain maximal functions and as a result brings in some classical harmonic analysis that arises so often in the first part of this book.
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