Izabella Laba, Kelan Zhai
Nazarov, Peres and Volberg proved recently that the Favard length of the $n$-th iteration of the four-corner Cantor set is bounded from above by $n^{-c}$ for an appropriate $c$. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.
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This is an expository paper detailing some of the recent advances on the problem, with emphasis on the number-theoretic method developed in my paper with Bond and Volberg for rational product sets (arXiv:1109.1031).
The Favard length of a subset of the plane is defined as the average of its orthogonal projections. This quantity is related to the probabilistic Buffon needle problem; that is, the Favard length of a set is proportional to the probability that a needle or a line that is dropped at random onto the set will intersect the set. If instead of dropping lines onto a set, we drop fixed curves, then the associated Buffon curve probability is proportional to the so-called Favard curve length. As we show in our companion paper, a Besicovitch generalized projection theorem still holds in the setting wher…
Favard length of non-homogeneous random disc-like Cantor sets
本文研究了非齐次随机四角康托尔集的Favard长度衰减，给出了尖锐估计，证明了Mattila下界的尖锐性，并构造了正解析容量但零Favard长度的新例子。
In this article, we consider the concept of the decay of the Favard length of $\varepsilon$-neighborhoods of purely unrectifiable sets. We construct non-self-similar Cantor sets for which the Favard length decays arbitrarily with respect to $\varepsilon$.
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。
We improve a special case of the Lam-Leung lower bound on the number of elements in a vanishing sum of $N$-th roots of unity. Using this result, we extend the Favard length estimates due to Bond, Łaba, and Volberg to a new class of rational product Cantor sets in $\mathbb{R}^2$.
Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres …
In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by $4$ disjoint subdisks (with equal distance to each other) of radius $1/4$ inside and tangent to the unit disk. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set ${\cal D}(ω)$. Let ${\cal D}_n$ be the $n$-th generation in the construction, which is comparable to the $4^{-n}$-neighborhood of ${\cal D}$. We are interested in the decay rate of the Favard le…
Improved Power Laws for the Favard Length Problem in All Dimensions
该研究针对全维度下纯1-不可求长有理乘积康托集的Favard长度小邻域建立定量上界，通过组合多种方法改进幂律，得到经典四角康托集及特定基下平面例子的相关指数界。
Let $S_\infty=A_\infty\times B_\infty$ be a self-similar product Cantor set in the complex plane, defined via $S_\infty=\bigcup_{j=1}^L T_j(S_\infty)$, where $T_j:\C\to\C$ have the form $T_j(z)=\frac1{L}z+z_j$ and $\{z_1,...,z_L\}=A+iB$ for some $A,B\subset\rr$ with $|A|,|B|>1$ and $|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighbourhood of $S_\infty$, or equivalently (up to constants), its $N$-th Cantor iteration. We are interested in the asymptotic behaviour as $N\to\infty$ of the {\it Favard length} of $S_N$, defined as the average (with respect to direction) length of its 1-dimensional projecti…
Transversal families of nonlinear projections and generalizations of Favard length
本文证明多类非线性投影满足横截性条件，并据此建立 Favard 曲线长度、Favard 曲面长度及径向可见性度量衰减的定量下界，还简化了四角 Cantor 集相关结果的证明。
Structure of sets with nearly maximal Favard length
本文研究平面有限长可测集 Favard 长度接近等长线段最大值时的几何结构，证明其除小长度集合外可由 ε-Lipschitz 图像覆盖，并给出 ε=Cδ^{1/70} 的多项式定量界。
Geometric Bounds for Favard Length
本文发展新的几何技术估计Favard长度，证明Hausdorff维数与邻域Favard长度衰减率的关系，并证明自相似集各代Favard长度序列的凸性，从而为分形集合提供Favard长度下界。
Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n = \Cant_n \times \Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\K_n$ is essentially the average length of the projections of $\K_n$, also known as the Favard length of $\K_n$. A classical theorem of Besicovitch implies that the Favard length of $\K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c…
In this paper we get an estimate of Favard length of an arbitrary neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of smaller disc onto the unit disc we can generate a self-similar Cantor set $G$. Then $\G=\bigcap_n\G_n$. One may then ask the rate at which the Favard length - the average over all directions of the length of the orthogonal projection onto a line in that direction - of these sets $\G_n$ decays to zero as a function of $n$. The quantitative results for the Favard length problem wer…
Favard length and generalized projections
该研究建立了广义投影与正交投影的局部可比性，推导了法瓦尔长度上界向非线性投影族的推广结论，还得到了纯不可约自相似1-集对应的圆并集及曲线并集的测度估计。
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 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.
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.
Let C be a Cantor set. For a real number t let C+t be the translate of C by t, We say two real numbers s,t are equivalent if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets determined by similarities with one fixed positive contraction ratio. For this class of Cantor set, we show that an "initial segment" of the intersection of C and C+t is a self-similar set with contraction ratios that are powers of the contraction ratio used to describe C as a self- similar set if and only if t is equivalent to a rational number. Our results …
The Buffon's needle problem for random planar disk-like Cantor sets
本文研究了有限正1-Hausdorff测度自相似Cantor集的第三种随机模型，证明了Buffon针落在其δ-邻域的概率衰减率为c/log(1/δ)，印证了任何合理的正有限长度随机Cantor集均具有该衰减阶的直觉。
Open intervals in sums and products of Cantor sets
本文通过新论证证明足够多个任意中心康托集的和与积可产生大的开区间，并将讨论推广到其 $C^1$ 像，为 Astels 的厚度结果提供新视角，同时处理康托集幂的和与积问题。
Cantor sets in higher dimension I: Criterion for stable intersections
本文在高维空间中研究Cantor集的稳定相交判据，并提出构造显式稳定相交Cantor集的方法，适用于Hausdorff维度极小的复杂情形。
We show that the Cantorvals connected with the geometric Cantor sets are not achievement sets of any series. However many of them are attractors of IFS consisting of affine functions.
These informal notes briefly discuss various aspects of Cantor sets.
In this paper we discuss several variations and generalizations of the Cantor set and study some of their properties. Also for each of those generalizations a Cantor-like function can be constructed from the set. We will discuss briefly the possible construction of those functions.
In these informal notes, we continue to explore p-adic versions of Heisenberg groups and some of their variants, including the structure of the corresponding Cantor sets.
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.
These informal notes deal with p-adic versions of Heisenberg groups and related matters.
Here we look at some related constructions of solenoids, and mappings associated to them.
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.
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.
We give lower bounds for the Hausdorff dimensions of some model Furstenberg sets.
Characterization of the algebraic difference of special affine Cantor sets
本文研究了由参数l,r,p∈N生成的自相似康托集S-康托集，分析了其代数差的结构特征，揭示了其可能为区间、康托集、L-康托瓦尔、R-康托瓦尔或M-康托瓦尔等形式。
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.
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.
We show that there exist $(d-1)$ - Ahlfors regular compact sets $E \subset \mathbb{R}^{d}, d\geq 2$ such that for any $t< d-1$, we have \[ \sup_T \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^t}<\infty \] where the supremum is over all tubes $T$ with width $w(T) >0$. This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.
The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the richness, the peculiarities and the generalities that has $C$ and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previous paper.
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…
The aim of this paper is to estimate the $L^2$-norms of vector-valued Riesz transforms $R_ν^s$ and the norms of Riesz operators on Cantor sets in $\R^d$, as well as to study the distribution of values of $R_ν^s$. Namely, we show that this distribution is "uniform" in the following sense. The values of $|R_ν^s|^2$ which are comparable with its average value are attended on a "big" portion of a Cantor set. We apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where Riesz transform is large, and for the corresponding Riesz capacities. …
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。
Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections
本文证明了对于正则Cantor集，上盒维数之和小于d时可被小变形分离，且此约束是最佳的。通过构造具有任意小厚度的正则Cantor集对，展示了C^{1+α}稳定交集的条件。
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 non-linear Roth theorem for thick Cantor sets
本文提出了一种非线性罗思定理，用于厚坎托集中的三项等差数列存在性证明。
Given a Cantor-type subset $Ω$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $Ω$ and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small $(d+1)$-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of $Ω$, and extends to higher dimensions an analogous planar result of Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set $Ω$ is unbounded …
We show that, for some Cantor sets in R^d, the capacity g_s associated to the s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to show that, when s is positive and non integer, they are comparable for all compact sets in R^d. We also discuss other open questions in the area.
It is well known that n x n upper-triangular real matrices with 1's on the diagonal form a nilpotent Lie group with an interesting family of non-isotropic dilations and corresponding geometry, as in [9]. Here we look at p-adic versions of this, and related matters.
Integer Cantor Sets: Arithmetic Combinatorial Properties
本文研究整数康托集的算术组合性质，探讨了交集性质、均匀分布、多项式均值遍历定理及度量对称相关性等核心问题。
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which includes zero. When it is absolutely summable the subsum set is one of the following: a finite union of (nontrivial) compact intervals, a Cantor set, or a "symmetric Cantorval".
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.
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。
For any $α\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ such that the associated natural measure $μ$ obeys the restriction estimate $\| \widehat{f dμ} \|_{p} \leq C_p \| f \|_{L^2(μ)}$ for all $p>2d/α$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $α=d/k$ with $k\in\mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $Λ(p)$ sets.
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。
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 …
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.
Let $C(a ),C(b)\subset \lbrack 0,1]$ be the central Cantor sets generated by sequences $ a,b \in (0,1)^{\mathbb{N}}$. The first main result of the paper gives a necessary and a sufficient condition for sequences $a$ and $b$ which inform when $C(a )-C(b)$ is equal to $[-1,1]$ or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets which algebraic difference is an interval may be arbitrarily small. We also show that there are sets $C(a)$ and $C(b)$ with the Hausdorff dimension equal to $0$ such …
We consider the self-similar structure of the class of generalized Cantor sets $$Γ_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_nβ^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<β<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $Γ_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.
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 $K$.
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 $|ξ| \to \infty$.
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.
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