Projection theorems for intermediate dimensions

Abstract

Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E \subset\mathbb{R}^n onto almost all m -dimensional subspaces depend only on m and E , that is, they are almost surely independent of the choice of subspace. Our approach is based on that are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the box dimensions of the projections of a set to the Hausdorff dimension of the set.

The fractal structure of elliptical polynomial spirals

“…Intuitively, the m-dimensional profile may be thought of as the dimension of an object when viewed from an m-dimensional viewpoint. In favour of brevity we omit a thorough introduction to dimension profiles, which may be found in [1,2,5,6], since the sole property we require is their relationship to fractional Brownian images [1,5,22]. In the following lemma, we bound the 2α-profiles of S p,q , denoted dim 2α θ S p,q , by a quantity strictly less than the dimension for all θ > 0, see Figure 6.…”

“…Intermediate dimensions have already seen surprising applications and properties, despite their recent introduction. For example, they have been used to establish relationships between the Hausdorff dimension of a set and the typical box dimension of fractional Brownian images [1] or orthogonal projections [2]. Other notable works include [16].…”

“…Figure 2. A plot of dim θ S p,q (y-axis) against θ (x-axis) for θ ∈ [0, 1] and dim θ−1 A S p,q against θ for θ ∈ [1,2]. In this example, p = 0.1 and q = 0.8.…”

On the intermediate dimensions of Bedford-McMullen carpets

“…As θ → 0, the θ-intermediate dimension gives more insight into which scales are used in the optimal cover to reach the Hausdorff dimension. Intermediate dimensions can also be formulated using capacity theoretic methods and may be used to relate the box dimensions of the projections of a set to the Hausdorff dimension of the set, see [2,3]. A similar concept of dimension interpolation between the upper box dimension and the (quasi-)Assouad dimension, called the Assouad spectrum was initiated in [8].…”

Fractal geometry of Bedford-McMullen carpets

“…Establishing continuity at 0 for particular examples turns out to be a key problem. For example, if the intermediate dimensions are continuous at 0, then strong applications can be derived concerning the box dimensions of projections and images under stochastic processes, see[B20,BFF19].…”