The distance set problem

The distance set problem is a notorious problem in geometric measure theory. It stems from a paper of Kenneth Falconer from 1985 and asks for the relationship between a subset of Euclidean and its distance set. The distance set associated with a given set is simply the set of all distances realised between pairs of points in the set. For example, if the dimension of a Borel set in d-dimensional Euclidean space is at least d/2, then it is conjectured that the dimension of the distance set should be 1 (which is maximal, since the distance set is a set of real numbers). This question is usually posed for Hausdorff dimension, but equally well applies to packing, box, Assouad dimension etc. I solved the Assouad dimension of the distance set problem for subsets of the plane here: I have used Fourier analytic tools to study the distance set problem: I have also used ergodic theoretic tools, such as CP-chains, to study the distance set problem. For example, this approach was used to obtain partial results on the Assouad dimension problem here: The ergodic theoretic approach is very useful when the underlying fractal is dynamically defined. I used this approach to solve the Hausdorff dimension version of the distance problem for certain self-affine and self-conformal sets here: I gave a talk about the ergodic theory approach to the distance set problem at a one day ergodic theory meeting in QMUL in 2015. The slides are here.

I gave an Analysis Seminar about the general solution to the Assouad dimension problem in St Andrews in 2020. Here is the video.