Related papers: Exact formula on upper box dimension of generic H\…

The $\alpha$-Weierstrass function is defined as $W_g^{\alpha,b}(x) = \sum_{k=0}^{\infty} b^{-\alpha k} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $\alpha$-Weierstrass function, we prove that the upper…

Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting…

For the spherical mean operators $\mathcal{A}_t$ in $\mathbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef =\sup_{t\in E} |\mathcal{A}_t f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising…

Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of…

In this paper, we investigate the boundary H\"{o}lder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic…

We first obtain the dimension formulas for the spaces of holomorphic modular forms with character for the Fricke group $\Gamma_0^+(N)$, then that for $\Gamma_0^*(N)$ with all Atkin-Lehner involutions added in a particular case.

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\text{Gr}(f))$ of the graph of a real-valued H\"older or Sobolev function $f$ defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds…

In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show…

Let X \subset R be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e.the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the…

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$.…

In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal sets $\mathbb{R}^{\alpha}$ $(0<\alpha\leq 1)$. Some properties of this new class are…

Let $ E $ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between…

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…

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any…

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$…

We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given…

Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there…

Let $\mbox{$\cal F$}\subseteq 2^{[n]}$ be a fixed family of subsets. Let $D(\mbox{$\cal F$})$ stand for the following set of Hamming distances: $$ D(\mbox{$\cal F$}):=\{d_H(F,G):~ F, G\in \mbox{$\cal F$},\ F\neq G\}. $$ $\mbox{$\cal F$}$ is…

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion…

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$…