Carlos A. Cabrelli, Kathryn E. Hare, Ursula M. Molter
In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.
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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。
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。
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additi…
Fractal uncertainty for discrete 2D Cantor sets
本文证明了离散二维Cantor集具有分形不确定性原理当且仅当其不含正交直线，关键运用了Lang猜想的定量形式，回答了Dyatlov的问题并应用于开放量子映射。
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.
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 $\mathbb{R}$.
In this paper, we study $C^ζ$-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the $C^ζ$-calculus on the generalized Cantor sets known as middle-$ξ$ Cantor sets. We have suggested a calculus on the middle-$ξ$ Cantor sets for different values of $ξ$ with $0<ξ<1$. Differential equations on the middle-$ξ$ Cantor sets have been solved, and we …
Infinite constant gap length trees in products of thick Cantor sets
本文证明充分厚的 Cantor 集乘积可在平面上生成相邻顶点间距为常数的树，且该距离取值集合具有非空内部，并允许树为可数无限，扩展了分形集中的模式研究。
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…
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。
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。
In this paper, we discussed the non-local derivative on the fractal Cantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared and related physical models are suggested.
The largest slice of fractal percolation
本文确定了平面分形渗流包含 $k$ 个共线点的维数阈值，并研究了临界维数下最大线性切片的广义豪斯多夫测度大小。
Sharp Favard length of random Cantor sets
研究平面一维随机分形邻域的Favard长度，证明其与log^{-1}(1/r)同阶，并揭示不同随机分形中衰减率的多样性。
In this article, we introduce a notion of size for sets called thickness that can be used to guarantee that two Cantor sets intersect (the Gap Lemma), and show a connection among Thickness, Schmidt Games and Patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions.
The article is devoted to one infinite parametric class of continuous functions with complicated local structure. In the article differential, integral, self-affine and other properties of functions, that their argument is represented by the alternating Cantor series are investigated. The functional equations systems, that investigating functions are unique solution of these systems in the class of determined and bounded on $[0;1]$ functions are indicated. The investigation was represented in seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine…
A new, extended nonlinear framework of the ordinary real analysis incorporating a novel concept of {\em duality structure} and its applications into various nonlinear dynamical problems is presented. The duality structure is an asymptotic property that should affect the late time asymptotic behaviour of a nonlinear dynamical system in a nontrivial way leading naturally to signatures generic to a complex system. We argue that the present formalism would offer a natural framework to understand the abundance of complex systems in natural, biological, financial and related problems. We show that t…
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 …
We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermor…
Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis
针对Hausdorff维数与上盒维数匹配的分形，提出在内部构建具有可控自相似性的拟Cantor子集的新方法，用于在经典质量传递原理失效时估计相交集合的维数下界，并应用于计算缺项小波级数的递增多重分形谱。
One-dimensional wave equations defined by fractal Laplacians
本文研究了一维由一类分形拉普拉斯算子定义的波动方程，通过自相似身份离散化方程并利用有限元和中心差分法得到弱解的数值近似，证明了数值解收敛于弱解并获得收敛率估计。
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