Scaling by 5 on a 1/4-Cantor Measure

Moments of the weighted Cantor measures

Based on the seminal work of Hutchinson, we investigate properties of {\em $α$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $α$ is a vector of nonnegative weights summing to $1$, and the corresponding weighted Cantor measure $μ^α$ is the unique Borel probability measure on $[0,1]$ satisfying $ μ^α(E) = \sum_{ n=0 }^{N-1} α_nμ^α( φ_n^{-1}(E) )$ where $φ_n: x\mapsto (x+n)/N$. In Sections 1 and 2 we examine several general properties of the measure $μ^α$ and the associated Legendre polynomials in $L_{μ^α}^2[0,1]$. In Section 3, we (1) compute the Lapl…

Cantor-Moran测度的谱性与Fuglede猜想

The spectrality of Cantor-Moran measure and Fuglede's Conjecture

本文引入双点条件集刻画Cantor-Moran测度的等正性，证明奇异连续情形均为谱测度，并研究绝对连续情形下等正性与支撑平铺的等价关系，推进Fuglede猜想。

About the multifractal nature of Cantor's bijection

In this note, we investigate the regularity of Cantor's one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by showing that its Hölder regularity lies between 0.35 and 0.72 almost everywhere (with respect to the Lebesgue measure).

Spectral property of Cantor measures with consecutive digits

We consider equally-weighted Cantor measures $μ_{q,b}$ arising from iterated function systems of the form ${b^{-1}(x+i)}$, $i=0,1,...,q-1$, where $q<b$. We classify the $(q,b)$ so that they have infinitely many mutually orthogonal exponentials in $L^2(μ_{q,b})$. In particular, if $q$ divides $b$, the measures have a complete orthogonal exponential system and hence spectral measures. We then characterize all the maximal orthogonal sets $Λ$ when $q$ divides $b$ via a maximal mapping on the $q-$adic tree in which all elements in $Λ$ are represented uniquely in finite $b-$adic expansions and we ca…

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also e…

Cantor sets of low density and Lipschitz functions on $C^1$ curves

We characterize the functions $f\colon [0,1] \longrightarrow [0,1]$ for which there exists a measurable set $C\subseteq [0,1]$ of positive measure satisfying $\frac{|C\cap I|}{|I|}<f(|I|)$ for any nontrivial interval $I \subseteq [0,1]$. As an application, we prove that on any $C^1$ curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant.

Measures on Cantor sets: the good, the ugly, the bad

We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the…

谱特征子空间与Cantor测度的树结构

Spectral Eigen-subspace and Tree Structure for a Cantor Measure

本文研究中间四分之一Cantor测度$\mu_4$的Fourier基构造，证明满足$E(\Lambda)$及其奇数缩放均为基的$\Lambda$具有连续统基数，并给出显式构造方法。

On Spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem

In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} μ= μ(N_n,B_n) =& δ_{\frac{1}{N_1}B_1}\astδ_{\frac{1}{N_1N_2}B_2}\ast δ_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& μ_n\astμ_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(Λ) = \{e^{2πi λx}:λ\inΛ\}$ for $L^2(μ)$, where $Λ$ is obtained from suitably modifying $L_n$. Here, $μ_n$ is the convolution of the first $n$ Dirac measures …

广义富格莱德猜想对一类康托-莫兰测度成立

The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures

针对由大于1的整数序列和连续数字集序列定义的康托-莫兰测度，证明其$L^2$空间具有指数正交基当且仅当存在博雷尔概率测度使卷积等于勒贝格测度，从而验证广义富格莱德猜想成立，并得到与积分平铺的等价性。

The Feichtinger Conjecture for Exponentials

The Feichtinger conjecture for exponentials asserts that the following property holds for every fat Cantor subset B of the circle group: the set of restrictions to B of exponential functions can be covered by Riesz sets. In their seminal paper on the Kadison-Singer problem, Bourgain and Tzafriri proved that this property holds if the characteristic function of B has Sobolev regularity. Their probability based proof does not explicitly construct a Riesz cover. They also showed how to construct fat Cantor sets whose characteristic functions have Sobolev regularity. However, these fat Cantor sets…