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 an expository paper on the general idea, which is probably a good place to start:

► An invitation to dimension interpolation, arXiv
        submitted

I have also written survey articles for proceedings of Fractal Geometry and Stochastics VI (2018) and Fractal Geometry and Harmonic Analysis (Banff, 2024):

► Interpolating between dimensions, arXiv
        Proceedings of Fractal Geometry and Stochastics VI, Birkhäuser, Progress in Probability, 2021.

► Applications of dimension interpolation to orthogonal projections, arXiv
        Research in the Mathematical Science, 12, (2025), 23 pp.

The main concepts in dimension interpolation (so far) were introduced across three papers. The paper in which we introduced the Assouad spectrum is here:

► New dimension spectra: finer information on scaling and homogeneity (with H. Yu), arXiv
        Advances in Mathematics, 329, (2018), 273-328.

The paper in which we introduced the intermediate dimensions is here:

► Intermediate dimensions (with K. J. Falconer & T. Kempton), arXiv
        Mathematische Zeitschrift, 296, (2020), 813-830.

The paper in which we introduced the Fourier spectrum is here:

► The Fourier spectrum and sumset type problems, arXiv
        Mathematische Annalen (to appear).

I developed a discrete analogue of the Fourier spectrum in the setting of vector spaces over finite fields here:

► L^p averages of the Fourier transform in finite fields, arXiv
        submitted.

Other work on these notions include the following:

► A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra (with L. Stuart), arXiv
        Bulletin of the American Mathematical Society (to appear).

► The fractal structure of elliptical polynomial spirals (with S. A. Burrell and K. J. Falconer), arXiv
        Monatshefte für Mathematik, 199, (2022), 1-22.

► Projection theorems for intermediate dimensions (S. A. Burrell & K. J. Falconer), arXiv
        Journal of Fractal Geometry, 8, (2021), 95-116.

► The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra (with K. E. Hare, K. G. Hare, S. Troscheit & H. Yu), arXiv
        Annales Academiæ Scientiarum Fennicæ Mathematica, 44, (2019), 379-387.

► Assouad type spectra for some fractal families (with H. Yu), arXiv
        Indiana University Mathematics Journal, 67, (2018), 2005-2043.

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.