On the sum of squares of middle-third Cantor set

Interiors of continuous images of the middle-third Cantor set

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial _{x}f$, $\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in (C\times C)\cap U$ such that \begin{equation*} 1<\left\vert \frac{\partial _{x}f|_{(x_{0},y_{0})}}{\partial _{y}f|_{(x_{0},y_{0})}}\right\vert <3\text{ or }1<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <3, \end{equation*} then $f…

Cantor集上的平均Fourier估计与二进逼近

Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

本文通过平均Fourier估计和二进逼近方法，研究了Cantor集上二进逼近的零一律，改进了Velani猜想的已知结果。

幂和，以及中间三分之一 Cantor 集元素的乘积

Sums of powers, and products of elements of the middle third Cantor set

本文提出一种新的初等动力学技术，统一处理中间三分之一 Cantor 集元素的幂和与幂积的测度及开区间问题，克服了已有方法在幂次增大时的困难。

Intersections of middle-$α$ Cantor sets with a fixed translation

For $λ\in(0,1/3]$ let $C_λ$ be the middle-$(1-2λ)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $Λ(t)$, which reveals a dimensional variation principle. Furthermore, for any $β\in[0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing d…

A note on dyadic approximation in Cantor's set

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, $K$, for approximation functions of the form $ψ_τ(n) = n^{-τ}$ ($τ\ge 0$). In particular, we show that for values of $τ$ beyond a certain threshold we have that almost no point in $K$ is dyadically $ψ_τ$-well approximable with respect to the natural probability measure on $K$. This refines a previous result in this direction obtained by the first, third, and fourth named authors (arXiv, 2020).

三分康托集上二进逼近的度量结果

Metric results for dyadic approximation on the middle-third Cantor set

研究三分康托集上二进逼近的度量性质，通过建立傅里叶系数估计，改进了Velani猜想的已知范围，将零测部分阈值降至约1.429，满测部分阈值升至约0.052。

Multiplication on uniform $λ$-Cantor sets

Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,÷$ (when $*=÷$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C÷C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] In this paper, we give a description of the topological structure and Lebesgue measure of $C\cdot C$. We indeed obtain corresponding results on the uniform $λ$-Cantor sets.

On the arithmetic sums of Cantor sets

Let $C_\la$ and $C_\ga$ be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions $d_\la$ and $d_\ga$, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having $\Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0$ where $C_\la + C_\ga$ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form $C_\la \times C_\ga$. We prove that for Lebesgue almost all directions $θ$ for which the projection is not one-to-one, the projection has zero $(d_…

Iterated functions and the Cantor set in one dimension

In this paper we consider the long-term behavior of points in ${\mathbb R}$ under iterations of continuous functions. We show that, given any Cantor set $Λ^*$ embedded in ${\mathbb R}$, there exists a continuous function $F^*:{\mathbb R}\to{\mathbb R}$ such that the points that are bounded under iterations of $F^*$ are just those points in $Λ^*$. In the course of this, we find a striking similarity between the way in which we construct the Cantor middle-thirds set, and the way in which we find the points bounded under iterations of certain continuous functions.

Self-similar subsets of the Cantor set

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.

The Minkowski sum of linear Cantor sets

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of …

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…

ℝᵈ中自相似集上τ-可逼近点的豪斯多夫维数

Hausdorff dimension of $τ$-approximable points on self-similar sets in $\mathbb R^d$

该研究针对ℝᵈ中满足开集条件的有限强不可约迭代函数系生成的非单点自相似集，确定了其上τ-可逼近点集与该自相似集交集的豪斯多夫维数，在一维中间三分康托集情形为Bugeaud-Durand猜想公式提供了支撑。