In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley, Salp, and Velani (2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured.
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).
Let $C$ be the middle third Cantor set and $μ$ be the $\frac{\log 2}{\log 3}$-dimensional Hausdorff measure restricted to $C$. In this paper we study approximations of elements of $C$ by dyadic rationals. Our main result implies that for $μ$ almost every $x\in C$ we have $$\#\left\{1\leq n\leq N:\left|x-\frac{p}{2^n}\right| \leq \frac{1}{n^{0.01}\cdot 2^{n}}\textrm{ for some }p\in\mathbb{N}\right\}\sim 2\sum_{n=1}^{N}n^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.
Let $C$ be the middle-third Cantor set. In this paper, we show that for every $x\in [0,4]$, there exist $x_1, x_2, x_3, x_4 \in C$ such that $$x= x_1^2+x_2^2+x_3^2+x_4^2,$$ which answers a question posed by Athreya, Reznick,and Tyson.
We prove upper and lower bounds for the Lebesgue measure of the set of products $xy$ with $x$ and $y$ in the middle-third Cantor set. Our method is inspired by Athreya, Reznick and Tyson, but a different subdivision of the Cantor set provides a more rapidly converging approximation formula.
We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supporting this heuristic.
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…
Dobiński set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $\mathcal{D}$ by means of the Mass Transference Principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $\mathcal{D}$.
Let $μ\geq 2$ be a real number and let $\Mcal(μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarní k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(μ)$ is equal to $2/μ$. We investigate the size of the intersection of $\Mcal(μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\Mcal(μ)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We especiall…
In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set, and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.
Given $b > 1$ and $y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\mathbb{N}\}$. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with `sufficiently regular' fractals $K\subset \mathbb{R}$ (that is, supporting measures $μ$ satisfying certain decay conditions). Furthermore, the int…
Cantor primes are primes p such that 1/p belongs to the middle-third Cantor set. One way to look at them is as containing the base-3 analogues of the famous Mersenne primes, which encompass all base-2 repunit primes, i.e., primes consisting of a contiguous sequence of 1's in base 2 and satisfying an equation of the form p + 1 = 2^q. The Cantor primes encompass all base-3 repunit primes satisfying an equation of the form 2p + 1 = 3^q, and I show that in general all Cantor primes > 3 satisfy a closely related equation of the form 2pK + 1 = 3^q, with the base-3 repunits being the special case K =…
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 …
The middle-third Cantor set C_3 is a fractal consisting of all the points in [0, 1] which have non-terminating base-3 representations involving only the digits 0 and 2. It is easily shown that the reciprocals of all prime numbers p > 3 satisfying an equation of the form 2p + 1 = 3^q belong to C_3. Such prime numbers have base-3 representations consisting of a contiguous sequence of 1's and are known as base-3 repunit primes. It is natural to ask whether all prime numbers with reciprocals in C_3 satisfy this equation. In this paper we show that the answer is no, but all primes with reciprocals …
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…
We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside …
Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $ε$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimens…
Intersections of Cantor sets with hyperbolas and continuous images
School of Sciences, Shanghai Institute of Technology（上海应用技术大学理学院）；School of Mathematics and Physics, School of Cryptology, Nanjing Institute of Technology（南京工程学院数学与物理学院、密码学院）；School of Science, Jiangsu Ocean University（江苏海洋大学理学院）
