Alex Rutar

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:

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Relevant Research Articles

  1. Assouad spectrum of Gatzouras–Lalley carpets
    With: Amlan Banaji, Jonathan M. Fraser, István Kolossváry
    Journal: Preprint (submitted)
    Links: pdfarxiv
  2. Tangents and pointwise Assouad dimension of invariant sets
    With: Antti Käenmäki
    Journal: Preprint (submitted)
    Links: pdfarxiv
  3. Interpolating with generalized Assouad dimensions
    With: Amlan Banaji, Sascha Troscheit
    Journal: Preprint (submitted)
    Links: pdfarxiv
  4. Assouad-type dimensions of overlapping self-affine sets
    With: Jonathan M. Fraser
    Journal: Ann. Fenn. Math. 49 (2024), 3–21
    Links: pdfdoiarxiv
  5. Attainable forms of Assouad spectra
    Journal: Indiana Univ. Math. J. (to appear)
    Links: pdfarxiv
  6. Attainable forms of intermediate dimensions
    With: Amlan Banaji
    Journal: Ann. Fenn. Math. 47 (2022), 939–960
    Links: pdfdoizblarxiv

Relevant Expository Articles

  1. Box versus packing dimensions via anti-Frostman measures
    Journal: Permanent Preprint
    Links: pdf
  2. Assouad dimension and self-similar sets satisfying the weak separation condition
    Journal: Permanent Preprint
    Links: pdf