To learn
https://en.wikipedia.org/wiki/Hamming_bound
https://en.wikipedia.org/wiki/Sphere_packing
https://en.wikipedia.org/wiki/Projective_plane
https://en.wikipedia.org/wiki/Fano_plane
https://www.ams.org/notices/200103/fea-tao.pdf
At first glance, Kakeya’s problem and Besicovitch’s resolution appear to be little more than mathematical curiosities. However, in the last three decades it has gradually been realized that this type of problem is connected to many other, seemingly unrelated, problems in number theory, geometric combinatorics, arithmetic combinatorics, oscillatory integrals, and even the analysis of dispersive and wave equations.
Applications to the Fourier Transform
Historically, the first applications of the Kakeya problem to analysis arose in the study of Fourier summation in the 1970s.
... Now suppose that f is a more general function, such as a function in the Lebesgue space Lp(Rn) . The Fourier inversion formula still holds true in the sense of distributions, but one is interested in more quantitative convergence statements. ...