Coarse geometry and dimension theory
Description
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.
- Amlan Banaji and I studied the lower box dimensions of infinitely-generated self-conformal sets. Primarily, we prove that there are only two mechanisms for the box dimension of the limit set to exist: either the Hausdorff dimension of the limit set dominates, or the box dimension of the set of fixed points exists. This gives (perhaps) the first example of a set of continued fraction expansions with restricted digit set for which the box dimension does not exist. More generally, we give an explicit asymptotic formula for the covering numbers of the attractor in terms of the Hausdorff dimension of the attractor and finer scaling properties of the set of fixed points.
- 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 the more usual 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.
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Relevant Research Articles
- A fractal local smoothing problem for the wave equation
With: David Beltran, Joris Roos, Andreas Seeger Journal: Preprint (submitted) Links: pdfarxiv - On the uniformity and size of microsets
With: Richárd Balka, Vilma Orgoványi Journal: Preprint (submitted) Links: pdfarxiv - Fibre stability for dominated self-affine sets
With: Roope Anttila Journal: Preprint (submitted) Links: pdfarxiv - Tangents and slices of self-affine carpets
With: Antti Käenmäki Journal: Preprint (submitted) Links: pdfarxiv - Regularity of non-autonomous self-similar sets
With: Antti Käenmäki Journal: Math. Proc. Camb. Philos. Soc. (to appear) Links: pdfarxiv - Lower box dimension of infinitely generated self-conformal sets
With: Amlan Banaji Journal: Preprint (submitted) Links: pdfarxiv - Assouad spectrum of Gatzouras–Lalley carpets
With: Amlan Banaji, Jonathan M. Fraser, István Kolossváry Journal: Preprint (submitted) Links: pdfarxiv - Tangents 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: pdfdoizblarxiv - Attainable forms of Assouad spectra
Journal: Indiana Univ. Math. J. 73 (2024), 1331–1356 Links: pdfdoizblarxiv - Attainable forms of intermediate dimensions
With: Amlan Banaji Journal: Ann. Fenn. Math. 47 (2022), 939–960 Links: pdfdoizblarxiv
Relevant Expository Articles
- Frostman’s lemma and subadditive functions
With: Vilma Orgoványi Journal: Permanent Preprint Links: pdf - Box versus packing dimensions via anti-Frostman measures
Journal: Permanent Preprint Links: pdf - Assouad dimension and self-similar sets satisfying the weak separation condition
Journal: Permanent Preprint Links: pdf