Dimension interpolation

One of the most beautiful and fundamental questions in fractal geometry is how to define the dimension of a fractal set or measure. There are many different ways to do this, each sensitive to a different geometric feature. Dimension interpolation is a new idea which seeks to understand the relationship between two given notions of dimension by viewing them as different facets of the same object: a continuum of dimensions varying from one to the other. There are three main ways to do this: the Assouad spectrum (introduced with Han Yu), intermediate dimensions (introduced with Kenneth Falconer and Tom Kempton) and the Fourier dimension spectrum (recently introduced). I wrote a survey paper on the general idea for the proceedings of Fractal Geometry and Stochastics VI, which is probably a good place to start: The paper in which we introduced the Assouad spectrum is here: The paper in which we introduced the intermediate dimensions is here: The paper in which we introduced the Fourier dimension spectrum is here: Other work on these notions include the following:

Plots of the Assouad spectrum (left) and intermediate dimensions (right) for an elliptical polynomial spiral. The Assouad spectrum typically has an increasing part and then a constant part, and the phase transition between these two regimes often has particular geometric significance. The smaller phase transition typically does not appear and was a surprise observation for this family of set.