Izabella Laba, Yang Wang
A probability measure in R^d is called a spectral measure if it has an orthonormal basis consisting of exponentials. In this paper we study spectral Cantor measures. We establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral Cantor measure. This extends the earlier results of Jorgensen-Pedersen and Strichartz.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
When is the fractal uncertainty principle for discrete Cantor sets most uncertain?
该研究给出离散康托集分形不确定性原理达到最不确定指数的充要条件，研究分布式谱对并完成循环群中的分类，还探讨了连续情形下的对应情况。
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…
A new type of deterministic Salem sets and its spectrality
本文为所有$0<s\le1$构造了Fourier维数与Hausdorff维数均为$s$的确定性Cantor集，利用Weil界实现快速Fourier衰减，并证明其自然测度具有指数正交基，首次给出$\mathbb{R}$上奇异Salem谱测度。
A characterization of uniqueness of purely atomic finite measures with central Cantor set range
该研究刻画了值域为中心康托集的纯原子有限测度的唯一性，推广到对称康托集并对康托尔集值域测度的恢复做初步探讨。
For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.
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 …
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture
本文研究一般谱 Cantor 测度的谱特征值问题，发现其与 Artin 原始根猜想密切相关，结果支持该猜想并印证 Minkowski 的观点。
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
We study a generalization of Morán's sum sets, obtaining information about the $h$-Hausdorff and $h$-packing measures of these sets and certain of their subsets.
We construct an iterated function system consisting of strictly increasing contractions $f,g\colon [0,1]\to [0,1]$ with $f([0,1])\cap g([0,1])=\emptyset$ and such that its attractor has positive Lebesgue measure.
We give an example of Cantor type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in Stahl-Totik sense.
We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.
In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Further, we achieve to construct a set, not just of positive measure, but of \textit{full measure} not containing any affi…
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided.
For $2\leq p<\infty$, $α'>2/p$, and $δ>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $α<α'$ for which the associated maximal operator is bounded from $L^p_δ(\mathbb{R})$ to $L^p(\mathbb{R})$. Maximal theorems for fractal measures on the line were previously obtained by Laba and Pramanik. The result here is weaker in that we are not able to obtain $L^p$ estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension $α>0$, and have no Fourier decay. The proof is based on a decoupling inequal…
We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2 \backslash K$ satisfies $G^p (x) \simeq \mathrm{dist} (x, K)^δ$ whenever $\mathrm{dist} (x, K) \le 1$ and $p$ is far from $K$.
Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere (with respect to their respective invariant measures) intersection.
Full measure universality for Cantor Sets
本文研究了康托集的全测度普遍性，证明在特定维度假设下，康托集不是全测度普遍的，并给出了相关结论。
In this article we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff-measure is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e. a Cantor set for which any translation invariant measure is either zero or non-sigma-finite, that enab…
A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em relative infinitesimals} and an {\em inversion} rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrisation invariant, is identified with the Cantor function associated with a Cantor set $\tilde C$ where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically …
In this survey article, we review some results and conjectures related to orthogonal polynomials on Cantor sets. The main purpose of this paper is to emphasize the role of equilibrium measures in order to have a general theory of sufficiently good measures (measures that behave similarly to measures which are in the Szegő class and the isospectral torus in the finite gap case) supported on totally disconnected subsets of $\mathbb{R}$. We present some open problems a number of which can be studied numerically.
In this paper we show that if $μ$ is any locally and uniformly $α$-dimensional measure supported on a $α$-quasi-regular set $E$, then $L^2(μ)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $μ_C,$ our result shows that there exists a countable set $Λ$ such that $\{e^{2πi t λ}\}_{λ\in Λ}$ is a frame for $L^2(μ_C)$ (i.e. the measure $μ_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L^2(μ_C)$.
Let mu(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize those r and s such that mu(r) and mu(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending mu(r) to mu(s)). The (possibly) weaker property of mu(r) and mu(s) being continuously reducible to each other is equivalent to a property of r and s called binomial equivalence. In this paper we define an algebraic property called "refinability" and show that, if r and s are r…
We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families $P_{n}(x)$ of orthogonal functions as functions of $n$ for fixed values of $x$.
We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal percolation measure. We study self-convolutions and Fourier decay of measures in our class, and present applications of these results to the restriction problem for fractal measures, and the connection between arithmetic structure and Fourier decay.
Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets
本文研究带等差数字集的 Cantor-Moran 测度，刻画有限层数字和集为整数砖的充要条件，并分析其与谱测度之间的必要、非充分及在附加条件下的等价关系。
Hausdorff measure of cartesian product of Cantor sets
本文研究了$1/3$ Cantor集$d$重笛卡尔积的Hausdorff测度界限，通过应用强开集条件定理并推广已有工作，计算了几个小整数$d$下更严格的上下界。
Rational points in Cantor sets and spectral eigenvalue problem for self-similar spectral measures
本文证明了当 $\gcd(p,q)=1$ 时，有限 $p$ 进制有理数与 Cantor 集 $K(q,A)$ 的交集有限当且仅当该集维数小于 1，并据此为自相似谱测度构造了在 $[0,+\infty)$ 稠密的谱特征值类，其对应特征子空间均为无限维。
For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_λ\}_{λ\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_λ>1$ for all $λ\in J$, under natural technical conditions we prove that the sum $K+C_λ$ has positive Lebesgue measure for almost all values of the parameter $λ$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.
On the spectra of Cantor measures
本文研究了线上的康托测度，考虑了收缩因子为$N^{-1}=p^{-α}$（其中$p$为正素数，$α$为正整数）且$m$个正整数位位于不同余类模$N$的测度。文章对这类测度$μ$的极大正交指数集进行了完整刻画，并证明了在极大正交集中的频率的第$n+1$位在前$n$位确定的情况下有$m$种可能值，从而建立了$m$-均匀根树的标记与极大正交频率集之间的对应关系。
For r in [0,1] let μ_r be the Bernoulli measure on the Cantor set given as the infinite power of the measure on the two-point set with weights r and 1-r. For r and s in [0,1] it is known that the measure μ_r is continuously reducible to μ_s (that is, there is a continuous map sending μ_r to μ_s) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin: Is it true that the product measures μ_r and μ_s are homeomorphic if and only if each is a conti…
We study the Fourier transforms $\widehatμ(ξ)$ of non-atomic Gibbs measures $μ$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehatμ(ξ) \to 0$ at a polynomial rate as $|ξ| \to \infty$.
Fractal uncertainty principle for random Cantor sets
本文研究了随机康托集的分形不确定性原理，通过不同的随机方法构建了维度在(0,2/3)内的随机康托集，并证明了在高概率下该原理成立。
Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure
本文针对中间四分之一康托测度，证明其类傅里叶级数的极大算子为弱型(1,1)，进而得到该级数几乎处处收敛，解决了Strichartz的长期开放问题。
Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0<a<1/2$, the measure $μ_a$ is supported on $C_a$, the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length $(1-2a)$, and iterating this process inductively on each of the remaining intervals. We investigate the convolutions $μ_a * (μ_b \circ S_λ^{-1})$, where $S_λ(x)=λx$ is a rescaling map. We prove that if the ratio $\log b/\log a$ is irrational and $λ\neq 0$, then \[ D(μ_a…
Sums of powers, and products of elements of the middle third Cantor set
本文提出一种新的初等动力学技术，统一处理中间三分之一 Cantor 集元素的幂和与幂积的测度及开区间问题，克服了已有方法在幂次增大时的困难。
Near-optimal restriction estimates for Cantor sets on the parabola
本文在抛物线上构造了维数为α的Salem康托集，证明其相关测度的Fourier变换满足L^p估计，且p>6/α的范围除端点外最优，方法结合了随机康托集限制估计与Fourier衰减技术。
Density spectrum of Cantor measure
本文研究了Cantor测度的逐点密度谱，通过引入涉及倍增映射的等价二进版本，结合开放动力系统熵与调谐概念，完整刻画了上下密度取值集合及其水平集的Hausdorff维数。
The centered maximal operator removes the non-concave Cantor part from the gradient
该研究针对d维空间中有界变差函数的中心Hardy–Littlewood极大函数正则性，证明非凹爆破点处的极大函数性质，并推导出满足特定条件时极大函数的BV正则性可提升至Sobolev正则性。
Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $Θ_*^s(μ,x)$ and upper $s$-density $Θ^{*s}(μ, x)$ of $μ$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ …
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。