Related papers: On the dimension theory of Okamoto's function
Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be…
We investigate the box-counting dimension of the image of a set $E \subset \mathbb{R}$ under a random multiplicative cascade function $f$. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the…
We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…
We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices…
In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as…
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal…
We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the…
We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the…
Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…
We compute the Hausdorff dimension of a two-dimensional Weierstrass function, related to lacunary (Hadamard gap) power series, that has no L\'evy area. This is done by interpreting it as a pullback attractor of a dynamical system based on…
We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on…
In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and size notions. In the case of H\"older functions, the authors have provided various…
For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of $\mathbb{R}^{2n}$ and for product sets.
Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a…
This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of…
A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the…
We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N --> oo through the primes or N=1 and k --> oo. We study the main and lower order terms in…
It is shown (Theorem A and its corollary) that if g is any nonconstant nonunivalent analytic function on a half-plane H and if D is either a half-plane or a smoothly bounded Jordan domain, then there is a function f on D for which f'(D)…
This article investigates the connectivity dimension of a graph. We introduce this concept in analogy to the metric dimension of a graph, providing a graph parameter that measures the heterogeneity of the connectivity structure of a graph.…
We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full…