单峰映射允许揉搓序列的维数和测度

本文证明了单峰映射的允许揉搓序列组成的集合在符号空间∑_2中的 Hausdorff 维数为1,1维 Hausdorff 测度为零。     

Sequences that omit a box (modulo 1)

Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in [0,1d⁾, we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S₁,…,Sd be strictly increasing in N. Define to be the set of x in [0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vr−ur→∞ and1+vr−ur?C#(Br). Kaufman has shown that is countable whenever S₁,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.     

Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in Rn akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978) [2], Dodson, Dickinson (2000) [18] and Beresnevich, Bernik, Kleinbock, Margulis (2002) [8]. In the case of planar curves, the complete Hausdorff dimension theory is developed.     

Concentration-cancellation phenomena for the velocity fields in incompressible fluid flows

A more intuitive sufficient condition is given for the concentration cancellation phenomena in 2- or 3-D incompressible fluid flows; that is, if the projection of concentration set of the weak-star defect measure associated with the approximate solution sequence onto space ℝ^π _x (n = 2, 3) is a set with Hausdorff dimension less than 1, then the weak-L ² limit of the approximate solution sequence is a classical weak solution of Euler equation. Using this condition, an example is given to elucidate concentration-cancellation phenomena.     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let

     

关于多项式小分数部分的一个注解

设f(x)为任意一实系数多项式,N.G.Moshchevitin在他的文章[8]中给出了集合{α∈R∶ limn→∞ infnlog n‖af(n)‖＞0}的Hausdorff维数的下界.在本文中,我们延用文[8]的方法并结合齐次Moran集的维数理论给出这个集合Hausdorff维数的精确值.     

Hausdorff dimension of fiber-coding sub-Sierpinski carpets

In this paper we study a class of subset of Sierpinski carpets for which the allowed digits in the expansions fall into each fiber set with a prescribed frequency. We calculate the Hausdorff and packing dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff and packing measures to be finite.     

On the Lower Bound of the Hausdorff Measure of the Koch Curve

This paper gives a lower bound of the Hausdorff measure of the Koch curve by means of the mass distribution principle.     

A localized Jarník–Besicovitch theorem

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x∈R:δx=δ}, where δ?1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x∈R:δx=f(x)}, where f is a continuous function. Our theorem applies to the study of the approximation exponents by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension.     

The Hausdorff dimension of sets related to the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.     

N指标d维广义Wiener过程象集的m项代数和的几个性质

设珮犠（狋）：犚犖＋ →犚犱是犖指标犱维广义Ｗｉｅｎｅｒ过程，对任意紧集犈１，…，犈犿犚犖＞ ，该文研究了犿项代数和珮犠（犈１）…珮犠（犈犿）的Ｈａｕｓｄｏｒｆｆ维数，Ｐａｃｋｉｎｇ维数和正的Ｌｅｂｅｓｇｕｅ测度及内点的存在性． 其结果包含并推广了布朗单的结果．     

R￣d中Hausdorff维数和packing维数的确定

本文目的在于建立确定Ｒ￣ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ和ｐａｃｋｉｎｇ维数Ｄｉｍ的两个命题（定理１和定理２），进而寻求Ｒ￣ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ与ｐａｃｋｉｎｇ维数Ｄｉｍ相等的条件；这使得我们能够引入分形测度的测度论定义。     

Hausdorff dimension and measure of a class of subsets of the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.     

On the Distribution Modulo 1 of Exponential Sequences

New quantitative results on the intersection of winning sets and the Hausdorff dimension of this intersection are obtained. An application to the problem on fractional parts of the sequence $\{ 2^n 3^m \alpha \}$ is given.     

稠密的s维分形集

对任意给定的0≤s≤1,本文构造Cantor型集E_s,使dim_H E_s=s,且E_s在[0,1]内稠密。     

The Hausdorff measure of a class of Sierpinski carpets

In this paper we obtain the exact value of the Hausdorff measure of a class of Sierpinski carpets with Hausdorff dimension no more than 1 and show the fact that the Hausdorff measure of such Sierpinski carpets can be determined by coverings which only consist of basic squares.     

限制零的m进制数集之Hausdorff测度

对于任意整数表示mkz的分数部分.给出了数集Fm的Hausdorff测度是Fm的HausdorfF维数.     

一类Sierpinski地毯的Hausdorff测度的估计

设Sr是压缩比为r(0.250≤r≤0.292)的Sierpinski地毯,该文证明了Sr的Hausdorff测度满足公式:21-s/2≤Hs(Sr)≤2s/2,其中s=-logr4.