Related papers: Assouad dimension and fractal geometry

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have…

It is shown that fractal dimension can be estimated seeking a solution of functional equation defined for areas of coverages of different scales. The method proposed is compared with widely known way to estimate fractal dimension via linear…

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish.…

We present a generalisation of the theory of iterated function systems and associated fractals to the setting of noncommutative geometry. Along the way, we discuss some ideas surrounding locally compact noncommutative metric spaces.

This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum, generalising previous results by Todd. We also study the multifractal spectrum of pointwise…

Orthogonal projections of the uniform measure on the Sierpinski triangle form a family of self similar measures with overlaps. The main result of this work is to make a connection between the dimension theory of these measures and the…

We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal…

We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(\mu)$ with measures $\mu$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics.…

Anomalous diffusion phenomena are ubiquitous in complex media, such as biological tissues. A wide class of sub-diffusive phenomena phenomena is described by the time-fractional diffusion equation. The paper investigates the case of…

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at…

Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63.…

Follow-up comment by the author: Theorem 2.2 in this paper is a special case of Theorems 1.1 and 4.1 in the article "Weighted thermodynamic formalism on subshifts and applications", Asian J. Math. 16 (2012), by J. Barral and D. J. Feng. In…

In this work, we examine the relationship between geometry and spectrum of regions with fractal boundary. The relationship is well-understood for fractal harps in one dimension, but largely open for fractal drums in larger dimensions. To…

We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.

Developing a robust generalization measure for the performance of machine learning models is an important and challenging task. A lot of recent research in the area focuses on the model decision boundary when predicting generalization. In…

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost…

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…

For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated…

Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…