From harmonic analysis to arithmetic combinatorics: a brief survey

The purpose of this note is to showcase a certain line of research that connects harmonic analysis, specifically restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent in-depth presentations of the various areas of research that we will discuss, see e.g. the references below. The emphasis here will be on highlighting the connections between these areas. Our starting point will be restriction theory in harmonic analysis on Euclidean spaces. The main theme of restriction theory, in this context, is the connection between the decay at infinity of the Fourier transforms of singular measures and the geometric properties of their support, including (but not necessarily limited to) curvature and dimensionality. For example, the Fourier transform of a measure supported on a hypersurface in R need not, in general, belong to any L with p < ∞, but there are positive results if the hypersurface in question is curved. A classic example is the restriction theory for the sphere, where a conjecture due to E.M. Stein asserts that the Fourier transform maps L∞(Sd−1) to L(R) for all q > 2d/(d− 1). This has been proved in dimension 2 (Fefferman-Stein, 1970), but remains open otherwise, despite the impressive and often groundbreaking work of Bourgain, Wolff, Tao, Christ, and others. We recommend [8] for a thorough survey of restriction theory for the sphere and other curved hypersurfaces. Restriction-type estimates have been immensely useful in PDE theory – in fact much of the interest in the subject stems from PDE applications. Harmonic analysis techniques are used to prove spacetime Lpx,t or mixed norm LqtL p x estimates for solutions to linear PDE, and these in turn have become ∗Supported in part by an NSERC Discovery Grant.