Curvature, combinatorics, and the Fourier transform (Q2756772)

This Notices article is a readable and painless introduction to the use of combinatorial methods in harmonic analysis (and conversely, the use of harmonic analysis methods in combinatorics). While there are many problems lying at the intersection between the two fields, the author here has chosen to focus on those problems relating to convex bodies which may or may not have curvature. Specifically, he discusses the Erdős distance problem for translation-invariant metrics; the Fuglede conjecture concerning the tiling and spectral properties of convex bodies; the lattice point problem, counting the number of integer points in a large dilate of a fixed convex body; the Nikodým set counterexample to the naive two-dimensional generalization of the fundamental theorem of calculus; and classification of \(L^\infty\) strip projection multipliers on \(Z^2\). These problems all involve to some extent some combinatorial techniques, the use of the Fourier transform, and a key role is played by the curvature of a certain convex body.