On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

Metric Diophantine approximation on the middle-third Cantor set

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…

丢番图逼近中Dirichlet谱的一个改进上界

An Improved Upper Bound for the Dirichlet Spectrum in Diophantine Approximation

通过引入Cantor型集F_4^*并证明其对数厚度大于1，结合Cantor集的和集结果，将射线原点常数δ的上界改进至约0.94866。

Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.

Thickness theorems with partial derivatives

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under some checkable conditions that the continuous image of arbitrary self-similar sets with positive similarity ratios is a closed interval, a finite union of closed intervals or containing interior. Third, we prove an analogous Erdős-Straus conjecture on the middle-third Cantor set. Finally, we consider the solutions to the Diophantine equations on fractal sets. M…

由具有有限不同素因子数的有理数进行内在迪奥菲安逼近

Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。

Extrinsic Diophantine approximation on manifolds and fractals

Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for ev…

关于丢番图集类的和集与笛卡尔积的度量结果

Metric results on sumsets and Cartesian products of classes of Diophantine sets

受Erdős关于刘维尔数和积结果启发，研究指定无理性指数实数类和集的度量性质，证明其和集勒贝格测度下几乎覆盖全体实数，补集豪斯多夫维数小，还拓展到笛卡尔积维数、缺数字康托尔集填充维数等问题，推广了Erdős的方法。

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 deci…

Cantor-winning sets and their applications

We introduce and develop a class of \textit{Cantor-winning} sets that share the same amenable properties as the classical winning sets associated to Schmidt's $(α,β)$-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our framework. As applications of this phenomenon we answer several previously open questions, including some re…

Number theoretic applications of a class of Cantor series fractal functions, II

It is well known that all numbers that are normal of order $k$ in base $b$ are also normal of all orders less than $k$. Another basic fact is that every real number is normal in base $b$ if and only if it is simply normal in base $b^k$ for all $k$. This may be interpreted to mean that a number is normal in base $b$ if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the $Q$-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical resu…

On Approximation constants for Liouville numbers

We investigate some Diophantine approximation constants related to the simultaneous approximation of $(ζ,ζ^{2},\ldots,ζ^{k})$ for Liouville numbers $ζ$. For a certain class of Liouville numbers including the famous representative $\sum_{n\geq 1} 10^{-n!}$ and numbers in the Cantor set, we explicitly determine all approximation constants simultaneously for all $k\geq 1$.

传递算子与非均匀福克西安晶格中不良集的维度

Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices

研究非均匀福克西安晶格中通过抛物固定点的迪奥菲安逼近，利用鲁厄和博文的热力学方法计算不良点集的维度，提出几何良好逼近与有界部分商数的联系。