Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points

The dispersion of a point set in the unit square is defined to be the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the generators of the lattice. This framework proves so successful that we were unable to ask a question that we could not answer. We give necessary and sufficient conditions under which a lattice has finite dispersion…

连分数集合的多尺度性质

Multi-scale properties of continued fraction sets

本文综述了限制连分数展开数字来自非空子集D⊂N的实数集的维数理论，涵盖Hausdorff维数、盒维数等，并讨论相关开问题。

具有有界奇偶阶部分商的连分数的Hausdorff维数

Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients

研究奇数阶部分商为1的连分数集合的大小，证明两个此类集合的和与积均包含非空区间，并确定奇数阶部分商为1且偶数阶部分商以指数、超指数及一般正函数速率增长时集合的Hausdorff维数。

On the existence of numbers with matching continued fraction and decimal expansions

A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_δ$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, w…

A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions

We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…

Borel--Bernstein 和 Hirst 型定理对于欧几里得虚二次域上的最接近整数复连分数

Borel--Bernstein and Hirst-type Theorems for Nearest-Integer Complex Continued Fractions over Euclidean Imaginary Quadratic Fields

本文研究了五类欧几里得虚二次域上的最接近整数复连分数映射，证明了两个度量结果，扩展了Borel-Bernstein定理和Hirst型定理，应用于稀疏模式、收缩目标和几乎必然的Levy-和Khinchine型定律。