We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of Łaba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $δ$, on the mass of the measure $μ$ together with an upper bound, $M$ on the $L^q$ norm of its Fourier transform for some $q\in(2,3]$ depending on the parameters $δ$ and $M$.
Let $r_k(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$ which does not contain any arithmetic progressions (APs) of length $k$. In this paper, we give new upper and lower bounds for fractal dimensions of a set which does not contain $(k,ε)$-APs in terms of $r_k(N)$, where $N$ depends on $ε$. Here we say that a subset of real numbers does not contain $(k,ε)$-APs if we can not find any APs of length $k$ with gap difference $Δ$ in the $εΔ$-neighborhood of the set. More precisely, we show multi-dimensional cases of this result. As a corollary, we find equivalences between multi-dim…
In the paper, we introduce the generalized convex function on fractal sets of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen inequality and generalized Hermite-Hadamard inequality. Furthermore,some applications are given.
Although Clifford analysis is like complex analysis in many ways, there are obvious differences related to noncommutativity, and a few aspects of this are considered here.
By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.
In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.
This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.
Let $A$ be a compact set in $\Rp$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $μ^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes $$ \Is(μ):=\iint\Rk{x}{y}{s}dμ(y)dμ(x)$$ over all such probability measures. In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then $μ^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.
We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.
We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions.
In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We study in detail the operation in $\mathcal{L}^p$ spaces and in sets of continuous functions, in a different way to previous works of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals w…
We investigate the asymptotic behavior of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, with algebraically dependent contraction ratios. For $s$ bigger than the dimension of the set $A$, we prove the asymptotic behavior of the minimal $N$-point Riesz $s$-energy of $A$ along explicit subsequences, but we show that the general asymptotic behavior does not exist.
In the present article a new class $Υ$ of all sets represented in the following form is introduced: $$ \mathbb S_{(s,u)}\equiv\left\{x: x= Δ^{s}_{{\underbrace{u...u}_{α_1-1}} α_1{\underbrace{u...u}_{α_2 -1}}α_2 ...{\underbrace{u...u}_{ α_n -1}}α_n...}, α_n \in A_0, α_n \ne u, α_n \ne 0 \right\}, $$ where $2<s\in \mathbb N$ and $u\in A$ are fixed parameters, $A=\{0,1,\dots,s-1\}$, and $A_0=A \setminus \{0\}$. Topological, metric, and fractal properties of these sets are studied. The problem of belonging to these sets of normal numbers is investigated. The theorem on the calculation of the Hausd…
We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.
In this paper, we discussed the non-local derivative on the fractal Cantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared and related physical models are suggested.
In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $α,β\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{αβ}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $β$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $α$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dim…
A new, extended nonlinear framework of the ordinary real analysis incorporating a novel concept of {\em duality structure} and its applications into various nonlinear dynamical problems is presented. The duality structure is an asymptotic property that should affect the late time asymptotic behaviour of a nonlinear dynamical system in a nontrivial way leading naturally to signatures generic to a complex system. We argue that the present formalism would offer a natural framework to understand the abundance of complex systems in natural, biological, financial and related problems. We show that t…
Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-Hölder-$α$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(α, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-Hölder-$α$ function defined on the frac…
The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the salient underlying fractal features of the data set. Instead, we employ the discrete energy, a technique borrowed from geometric measure theory, to limit the number of points of a given data set that lie near a $k$-dimensional hyperplane, or, more generally, near a set of a given upper Minkowski dimension. Concrete motivations stemming from naturally arising …
We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Ha…
Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-Hölder-$α$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations g…
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additi…
The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in R^d (cf. [3]). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in [3] suggest that there is much more flexibility for the lower contents and dimensions. We show that this is indeed the case. There are sets whose lower S-dimension is strictly smaller than their lower Minkowski dimension. More precisely, given…
A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $ε_0>0$ such that for $x\in E$ and $0<r\leq {\rm diam}(E)$, $E\cap B(x,r)$ never stays $ε_0r$-close to a hyperplane in ${\Bbb R}^d$. Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self-similar sets, self-conformal sets in ${\Bbb R}^d$ (with $d\geq 2$) and self-affine sets in …
In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.
In recent years there has been much interest -and progress- in understanding projections of many concrete fractals sets and measures. The general goal is to be able to go beyond general results such as Marstrand's Theorem, and quantify the size of every projection - or at least every projection outside some very small set. This article surveys some of these results and the techniques that were developed to obtain them, focusing on linear projections of planar self-similar sets and measures.
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $δ$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques.
In this article, we introduce a notion of size for sets called thickness that can be used to guarantee that two Cantor sets intersect (the Gap Lemma), and show a connection among Thickness, Schmidt Games and Patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions.
We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, like the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.
An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the `most difficult location and scale' at which to cover the set and its difference from box dimension can be thought of as a first-level overall measure of how inhomogeneous the set is. For the next level of analysis, we develop a quantitative theory of inhomogeneity by considering the measure of the set of points around which the set exhibits a given level of inhomogeneity at a certain scale. For a set of examples, a family of $(\times …
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemerédi theorems for random discrete sets, we also consider the corresponding problem for sets of positive $ν$-measure, where $ν$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating t…
This is a book to be published in 2020 by Cambridge University Press (Tracts in Mathematics Series). It focuses on the Assouad dimension of sets and measures in Euclidean space, as well as many variants on the Assouad dimension, including the lower dimension and Assouad spectrum. It considers these notions in the setting of fractal geometry and topics covered include: weak tangents, self-similar sets, self-affine sets, Mandelbrot percolation, limit sets of Kleinian groups, orthogonal projections, distance sets, and Kakeya sets. It also considers applications of these ideas to other areas, incl…
