Coarse geometry and dimension theory
Coarse (or “fractal”) sets are sets which exhibit complex geometric properties at arbitrarily small scales. One common way to understand the fine scaling properties of a set is from the perspective of dimension theory, which attempts to assign and study various dimension-like invariants for sets, and to relate the properties of these dimensions to the geometry of the underlying set.
One family of dimensions which I have focused on are given by the Assouad-type dimensions, which are a family of dimensions which capture the “coarsest” scaling properties of a set. This area has been very active recently, with a large amount of interest in this field from various points of view, such as conformal geometry, embedding theory, and fractal geometry.
Another concept my research has focused on is the notion of dimension interpolation. In this scheme, one takes two commonly studied notions of dimension (for example, Hausdorff and box dimensions) and attempts to define a continuously parametrized family of dimensions which interpolate between the endpoints. Some examples include the (generalized) intermediate dimensions, the Assouad spectrum, the generalized Assouad dimensions, and the Fourier dimension spectrum
Here is a summary my work in this area:
- I have proven classification results for intermediate dimensions (with Amlan Banaji) and Assouad spectra.
- Antti Käenmäki and I introduced the pointwise Assouad dimensions, which is a localized version of the Assouad dimension, and studied it for a variety of iterated function system attractors. In this paper, we also provide a detailed study of some self-affine carpets, which can be thought of as complementary to my joint paper with Jonathan Fraser on the Assouad dimension of overlapping self-affine carpets with Jonathan Fraser. One common theme is the concept of a symbolic slice, first introduced in my joint work with Fraser.
- I have studied the generalized Assouad dimensions, both in a general setting and for sets associated with branching processes and iterated function systems, in a joint paper with Amlan Banaji and Sascha Troscheit.
- I have written an expository article giving two proofs of the fact that the packing dimension is countably stabilized upper box dimension (one using the anti-Frostman construction of Falconer–Fraser–Käenmäki and a complementary proof using dyadic pigeonholing).
- I have written an expository article giving a short proof of the Assouad dimension dichotomy result originally due to Fraser–Henderson–Olsen–Robinson.
Relevant Research Articles
- Assouad spectrum of Gatzouras–Lalley carpets
With: Amlan Banaji, Jonathan M. Fraser, István Kolossváry Journal: Preprint (submitted) Links: pdfarxiv
- Tangents and pointwise Assouad dimension of invariant sets
With: Antti Käenmäki Journal: Preprint (submitted) Links: pdfarxiv
- Interpolating with generalized Assouad dimensions
With: Amlan Banaji, Sascha Troscheit Journal: Preprint (submitted) Links: pdfarxiv
- Assouad-type dimensions of overlapping self-affine sets
With: Jonathan M. Fraser Journal: Ann. Fenn. Math. 49 (2024), 3–21 Links: pdfdoiarxiv
- Attainable forms of Assouad spectra
Journal: Indiana Univ. Math. J. (to appear) Links: pdfarxiv
- Attainable forms of intermediate dimensions
With: Amlan Banaji Journal: Ann. Fenn. Math. 47 (2022), 939–960 Links: pdfdoizblarxiv