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:
► A nonlinear projection theorem for Assouad dimension and applications, arXiv
Journal of the London Mathematical Society (to appear).
I have used Fourier analytic tools to study the distance set problem:
► The Fourier dimension spectrum and sumset type problems, arXiv
submitted.
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:
► Distance sets, orthogonal projections, and passing to weak tangents, arXiv
Israel Journal of Mathematics, 226, (2018), 851-875.
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:
► Scaling scenery of (×m,×n) invariant measures (with A. Ferguson & T. Sahlsten), arXiv
Advances in Mathematics, 268, (2015), 564-602.
► Micromeasure distributions and applications for conformally generated fractals (with M. Pollicott), arXiv
Mathematical Proceedings of the Cambridge Philosophical Society, 159, (2015), 547-566.
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.