Effective methods in geometric measure theory and fractal geometry originated in a 2018 paper of J. Lutz and N. Lutz on the Kakeya conjecture. Since then, mathematicians and theoretical computer scientists have applied these tools to prove new distance set bounds, partially resolve the Furstenberg set conjecture, establish sharp radial projection estimates, and much more. During the course of these achievements, this algorithmic approach has developed and matured into a theory with characteristic tricks, techniques, and lemmas.
This is the first in a series of \(N \geq 2\) talks on effective methods with the goal of laying down the foundations for a general fractal geometry audience. In this lecture, we will begin with the fundamentals: oracle machines, Kolmogorov complexity, the point-to-set principles, and symmetry of information. These basic objects and results are all we will need to prove a simple pinned distance set theorem for arbitrary norms, generalizing a result of Falconer.