Related papers: Generalised intermediate dimensions
We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs.…
In this paper, we answer a question of Nussbaum, Priyadarshi, and Lunel [Positive operators and Hausdorff dimension of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066.]. We also show that the Hausdorff dimension and box…
This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted…
We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key question is `what is the generic dimension of $f(X)$?' and we consider two different…
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…
In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized…
By extending the definition of boxicity, we extend a Helly-type result given by Danzer and Grumbaum on 2-piercings of family of boxes in $d$-dimensional Euclidian space by lowering the dimension of the boxes in the ambient space.
In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(\Phi )=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq \Phi(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\…
We investigate the continuity of Hausdorff dimension and box dimension (limit capacity) of non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomophisms through a Hopf bifurcation studied by Horita and Viana (see…
We introduce a family of piecewise isometries. This family is similar to the ones studied by Hooper and Schwartz. We prove that a renormalization scheme exists inside this family and compute the Hausdorff dimension of the discontinuity set.…
Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and…
This paper introduces two new abstract morphs for two $2$-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are…
We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…
This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in…
This is a revised compilation of the papers arXiv:1105.1554 and arXiv:1105.5823. We develop some of the ideas belonging to W.Schmidt and L.Summerer to define intermediate Diophantine exponents and split several transference inequalities…
We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while…
Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the…
We introduce the notion of scale to generalize and compare different invariants of metric spaces and their measures. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They moreover are…
An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot-Carath\'eodory metric. This improves the known lower bound, and answers a question of…
Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $\Phi$-badly approximable real vectors of $\mathbb{R}^m$ with respect to an approximation function $\Phi$, and determined its Hausdorff dimension for the…