Cai-Yun Ma, Yu-Feng Wu
We prove that for all $s\in(0,d)$ and $c\in (0,1)$ there exists a self-similar set $E\subset \mathbb{R}^d$ with Hausdorff dimension $s$ such that $\mathcal{H}^s(E)=c|E|^s$. This answers a question raised by Zhiying Wen[16].
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Using a recent result of Orponen (Invent. math. '21), we show that sets with plenty of big projections (PBP) admit an Analyst's Travelling Salesman Theorem. We then show that sets with PBP which are uniformly non-flat (or wiggly) have large Hausdorff dimension. We also obtain a corollary on analytic/Lipschitz harmonic capacities.
Through the use of a nonstandard version of Frostman's lemma, the notion of Hausdorff dimension is formulated in nonstandard euclidean space of arbitrary dimension. This allows for a nonstandard proof of the Kakeya conjecture in two dimensions, which can then be extended to the three-dimensional case.
Let $φ$ be a nonnegative integrable function on $(0,\infty)$. It is well-known that the Hausdorff operator $\mathcal H_φ$ generated by $φ$ is bounded on the real Hardy space $H^1(\mathbb R)$. The aim of this paper is to give the exact norm of $\mathcal H_φ$. More precisely, we prove that $$\|\mathcal H_φ\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty φ(t)dt.$$
For functions $f$ whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of $w(y):=|f(γ+iy)|$ for $y>0$ ($γ\leq1$ fixed).
Approximation via partial Hausdorff integrals on $H^1(\mathbb{R})$
本文通过部分霍尔德积分在$H^1(\mathbb{R})$范数下逼近函数，并将Pinos和Liflyand的结果从$L^p$空间扩展到$H^1$空间。
This paper shows $$ \sup_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{it(-Δ)^α}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}\ \ \text{under}\ \ \begin{cases} n\geq2;\\ α>\frac12; \frac{n}{2(n+1)}<s\leq\frac{n}{2} . \end{cases} $$
On the parabolic Hausdorff dimension of orthogonal projections
本文证明了Borel集A⊂ℝⁿ×ℝ在ℝⁿ×ℝ的通用m维线性子空间上的正交投影的抛物型Hausdorff维数的下界。
This paper is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved one-dimensional family of two-dimensional subspaces, the main result states that there exists $σ(s) > 1$ such that $\dim π_{V}(K) \geq σ(s)$ for almost all $V \in \mathcal{G}$. A similar result is obtained for some specific families of one-dimensional subspaces.
New bound on the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$
该研究将Wang-Zahl的三维Kakeya集研究与Katz-Zahl的平面刷论证结合，改进Guth-Zahl的三线性界，得到R⁴中Kakeya集Hausdorff维数的新下界3.073。
In this note, we show that the Hausdorff operator $H_Φ$ is unbounded on a large family of Quasi-Banach spaces, unless $H_Φ$ is a zero operator.
In the present paper, we characterize the nonnegative functions $φ$ for which the multi-parameter Hausdorff operator $\mathcal H_φ$ generated by $φ$ is bounded on the multi-parameter Hardy space $H^1(\mathbb R\times\cdots\times\mathbb R)$ or $L^p(\mathbb R^n)$, $p\in [1,\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \cite{Li}.
An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$
该研究改进了三维欧氏空间中Besicovitch集的Hausdorff维数下界至$5/2+ε_0$，通过分析满足Wolff公理的管并体积得到结果，还提出了Kakeya猜想的新近乎反例$SL_2$示例供后续研究。
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the formula.
The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $\mathcal H^n(X)$, is finite. Suppose further that the lower n-density of the measure $\mathcal H^n$ is positive, $\mathcal H^n$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $\mathcal H^n$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results…
An improved bound on the Hausdorff dimension of sticky Kakeya sets in $\mathbb{R}^4$
本文针对$\u211d^4$中的粘性Kakeya集，结合平面刷论证与粘性性质，将其Hausdorff维数下界从原有3.059提升至3.25，推进了粘性Kakeya猜想的相关研究。
For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of $\mathbb{R}^{2n}$ and for product sets.
Hausdorff and box dimension of self-affine set in non-Archimedean field
本文在局部紧非阿基米德域中建立奇异值分解理论，并在一般情形下计算自仿射集的盒维数与Hausdorff维数，且无需实数情形中的线性部分范数限制。
We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of $\mathbb{R}^n$ affect the sizes of the images of sets of Hausdorff dimension less than $n$. We measure the sizes of the image sets in terms of generalized Hausdorff measures.
Let $p\in(1,\infty)$, $q\in[1,\infty)$, $s\in\mathbb{R}$ and $τ\in[0, 1-\frac{1}{\max\{p,q\}}]$. In this paper, the authors establish the $φ$-transform characterizations of Besov-Hausdorff spaces $B{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ and Triebel-Lizorkin-Hausdorff spaces $F{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ ($q>1$); as applications, the authors then establish their embedding properties (which on $B{\dot H}_{p,q}^{s,τ}(\mathbb{R}^n)$ is also sharp), smooth atomic and molecular decomposition characterizations for suitable $τ$. Moreover, using their atomic and molecular decomposition characteriz…
These notes deal with metric spaces, Hausdorff measures and dimensions, Lipschitz mappings, and related topics. The reader is assumed to have some familiarity with basic analysis, which is also reviewed.
Hausdorff dimension of plane sections and general intersections
本文推广了平面截面维数公式，去除了正下密度假设，并应用于一般交集维数下界及例外旋转集维数估计。
We prove that for any $1 \le k<n$ and $s\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < α\le k$, if $B \subset {\mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${\mathbb R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $α$, then $\dim B \ge 2 α- k + \min(\dim E, 1)$, where $\dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well known Furstenberg-typ…
We show that the operations addition and multiplication on the set $C(Ω)$ of all real continuous functions on $Ω\subseteq\mathbb{R}^n$ can be extended to the set $\mathbb{H}(Ω)$ of all Hausdorff continuous interval functions on $Ω$ in such a way that the algebraic structure of $C(Ω)$ is preserved, namely, $\mathbb{H}(Ω)$ is a commutative ring with identity. The operations on $\mathbb{H}(Ω)$ are defined in three different but equivalent ways. This allow us to look at these operations from different points of view as well as to show that they are naturally associated with the Hausdorff continuou…
Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus
本文证明了环面上豪斯多夫维数大于1-ε的紧集必包含多项式构型，并由此推出相关多项式多重遍历平均发散集的维数严格小于1。
For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding vertices of $G$ are connected by an edge. We regard two such frameworks as equivalent if the specified distances are the same. We show that in a suitable sense the set of equivalences of such frameworks naturally embeds in ${\mathbb R}^m$ where $m$ is the number of "essential" edges of $G$. We prove that there exists a threshold $s_k<d$ such that if the Hausdor…
The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.
A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
本文证明在固定指标 p 时，Capacity Muckenhoupt 权类关于指标 δ 不满足自改进性质，补充了该类权性质的研究。
In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…
It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz graphs. We say that $G\subset \mathbb{R}^k\times \mathbb{R}^n$ is $Γ_k$-null if for every Lipschitz function $f:\mathbb{R}^k\to \mathbb{R}^n$ the set $\{t\in\mathbb{R}^k\,:\,(t,f(t))\in G\}$ has measure zero. We show that for every Borel set $E\subset \mathbb{R}^k\…
In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres.
We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + ξ_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $ξ_n\in\mathbb T^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.
A short note on the dimension of the CC-geodesic Kakeya sets in the first Heisenberg group
该研究针对第一Heisenberg群的CC测地线Kakeya集，修正了Kakeya猜想的自然预测，证明其精确下界为4，构造了满维零测集及不同曲率的低维零测Kakeya集，完善了相关维数理论。
We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $α$ with $k-1 < α\leq k$, then $\dim B \geq α+\dim E/(k+1)$, where $\dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $α$-Furstenberg set in the plane has Hausdorff dimension at least $α+ 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < α\leq k$, then $\dim B \geq α+(\dim E-(k-\lcei…
Hausdorff measure of zeros of polynomials
本文通过初等证明，确立了任意非平凡多变量多项式在指定大小立方体内零点的Hausdorff测度的精确普适界限，该界限仅依赖于立方体大小、维度及多项式次数。
The nonlinear Hausdorff-Young inequality
该研究证明了常数1的离散非线性豪斯多夫-杨不等式，通过极限论证得到连续空间的对应不等式，解决了Muscalu-Tao-Thiele一致性问题，其证明利用隐藏的希尔伯特空间结构将非线性问题转化为经典插值问题。
We show that the Szegő matrices, associated with Verblunsky coefficients $\{α_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^γ|α_n|^2 < \infty$ for some $γ\in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1 - γ$. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than $1-γ$. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.
Representation of quasi-periodic functions and Hausdorff-Young inequalities for Besicovitch almost periodic functions
本文研究了准周期函数的表示及其与贝西科维奇几乎周期函数的豪斯多夫-扬不等式的关系，通过建立Banach代数的同构关系，给出了简化的不等式证明并探讨了母函数的正则性。
Curly Kakeya: The Hausdorff dimension of sets containing circles or line segments in many directions
研究包含多个方向圆的集合的豪斯多夫维度下界，通过分析球面极点和经线的结构，证明该集合的维度至少为n-1。
Barry Simon conjectured in 2005 that the Szegő matrices, associated with Verblunsky coefficients $\{α_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^γ|α_n|^2 < \infty$ for some $γ\in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1 - γ$. Three of the authors recently proved this conjecture by employing a Prüfer variable approach that is analogous to work Christian Remling did on Schrödinger operators. This paper is a companion piece that presents a simple proof of a weak version of Simon's conjecture that is in the spirit o…
Marstrand's projection theorem from $1954$ states that if $K \subset \mathbb{R}^{3}$ is an analytic set, then, for $\mathcal{H}^{2}$ almost every $e \in S^{2}$, the orthogonal projection $π_{e}(K)$ of $K$ to the line spanned by $e$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. This paper contains the following sharper version of Marstrand's theorem. Let $V \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. Then, for $\mathcal{H}^{1}$ almost every $e \in S^{2} \cap V$, the projection $π_{e}(K)$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. For $0 \leq t < \…
Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements
本文证明 Kakeya 猜想在紧/Borel/任意集、线段/直线及大方向集等表述下等价，并通过加性补集建立 Hausdorff 维数与 packing 维数的对偶公式。
We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets (for this definition of thickness) with winning sets, give a lower bound for the Hausdorff dimension of the intersection of countably many of them, a result guaranteeing the presence of large patterns, and lower bounds for the Hausdorff dimension of a set in relationship with its thickness.
Hausdorff operators on weighted Bergman and Hardy spaces
本文研究上半平面加权Bergman空间与幂加权Hardy空间上的Hausdorff算子，给出其定义并探讨该算子实版本的相关应用，属于复分析与调和分析领域的算子理论研究。
Separating covering numbers and separating uniformity numbers of Hausdorff measures need large continuum
本文证明了在 $\mathfrak{c} = \aleph_2$ 的假设下，Hausdorff 测度的所有覆盖数和一致数各自相等，部分回答了文献[4]中的问题5.3和5.4。
Let $E=\limsup\limits_{n\to\infty}(g_n+ξ_n)$ be the random covering set on the torus $\mathbb{T}^d$, where $\{g_n\}$ is a sequence of ball-like sets and $ξ_n$ is a sequence of independent random variables uniformly distributed on $\T^d$. We prove that $E\cap F\neq\emptyset$ almost surely whenever $F\subset\mathbb{T}^d$ is an analytic set with Hausdorff dimension, $\dim_H(F)>d-α$, where $α$ is the almost sure Hausdorff dimension of $E$. Moreover, examples are given to show that the condition on $\dim_H(F)$ cannot be replaced by the packing dimension of $F$.
Multiversion of the Hausdorff--Young inequality
本文研究相关联合高斯向量上经Mehler变换或噪声算子作用后的多函数不等式，统一推广实复超压缩性，并得到Hausdorff–Young等不等式的多函数版本及含噪Brascamp–Lieb刻画。
For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $π^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$π^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $π^x$ preserves the Hausdorff dimension of $E$, namely whether $\dim_{\mathcal{H}}π^x(E)=\dim_{\mathcal{H}} E$. We develop a general framework to link $π^x(E)$, $x\in F$ and $π^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{R}^d$. In particular, whether $\dim_{\mathcal{H}}π^x(E)=\di…
Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $φ_λ: Δ_M φ_λ+ λφ_λ= 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $λ$. We show that $$H^{n-1}(\{ φ_λ=0\}) \leq C λ^α,$$ where $α>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from …
The purpose of this paper is to complete the proof of the following result. Let $0 < β\leq α< 1$ and $κ> 0$. Then, there exists $η> 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$, then $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. This paper is a sequel to the author's previous work from 2021 which, roughly speaking, established the same result with $\dim_{\mathrm{H}} (A + cB)$…
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