Besicovitch sets which were constrcted when Besicovitch was solving Kakeya problem, become the key issue to study some conjectures in harmonic analysis and PDEs. This conjecture has been found close relation with Bochner-Riesz conjecture, restriction conjecture, localized smoothing conjecture. These conjectures related to harmonic analysis, PDEs, number theory, geometric measure theory, and arithmetic combinatorics, etc. On the other hand, the methods developed to solve these conjectures played important roles in studying other famous conjectures, such as Vinogradov conjecture, Montgormery conjecture, Waring problem, Riemann-Zeta function, etc. We believe that all these conjectures might be different representation of a single core problem. This talk is based on localized smoothing conjecture, Bochner-Riesz conjecture, restriction conjecture, Kakeya conjecture, and introduce the related conjectures, their progress, and modern methods to study these conjectures (such as blinear method by Hormander-Fefferman-Cordoba, L^p orthogonal method by Wolff, Bourgain-Guth method, microlocal analysis and profile decomposition method, etc) and their roles in the fields of PDEs, mathematical physics, and number theory, etc.