In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $α,β\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{αβ}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $β$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $α$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{α+\fracβ{2} ; 2α+β-1\right\}$ for any $E\in F_{αβ}$. In particular we are able to extend previously known results to the ``endpoint'' $α=0$ case.
Kakeya and Furstenberg problems for sticky sets of tubes
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）
In this paper, we show that circular $(s,t)$-Furstenberg sets in $\mathbb R^2$ have Hausdorff dimension at least $$\max\{\frac{t}3+s,(2t+1)s-t\} \text{ for all $0<s,t\le 1$}.$$ This result extends the previous dimension estimates on circular Kakeya sets by Wolff.
We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar $(α,2α)$-Fursterberg sets. This provides a quantitative improvement to the $2α+ε$ bound of Héra-Shmerkin-Yavicoli. In particular, we show that every $1/2$-Furstenberg set has dimension at least $1 + 1/4536$.
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…
We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p \mod 1$ and $\times q \mod 1$, respectively, and if $\log p/\log q\notin \mathbb{Q}$, then for all real numbers $u$ and $v$, $$\dim_{\rm H}(uA+v)\cap B\le \max\{0,\dim_{\rm H}A+\dim_{\rm H}B-1\}.$$ We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on $\mathbb{R}$. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducibl…
Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}^{d}$, and let $π_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $μ$ is a compactly supported Radon measure on $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $μ(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|π_{V}μ\|_{L^{p}(V)}^{p} \, dγ_{d,n}(V) < \infty, \qquad 1 \leq p < \frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \leq s \leq d$, and every $0 < n < d$. Our …
This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.
我们发现，足够大的Hausdorff和傅立叶维数的$\mathbb{R}^n$的子集包含多项式模式的形式\begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*}其中$A_i$是真正的$n \times m$矩阵，$Q$是一个真正的多项式在$m$变量和$e_n = (0,\dots,0,1)$。
Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + ε, $$ where $ε= ε(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $μ$ is a Borel measure on $\mathbb{P}$ satisf…
We investigate the asymptotic behavior of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, with algebraically dependent contraction ratios. For $s$ bigger than the dimension of the set $A$, we prove the asymptotic behavior of the minimal $N$-point Riesz $s$-energy of $A$ along explicit subsequences, but we show that the general asymptotic behavior does not exist.
Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-Hölder-$α$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(α, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-Hölder-$α$ function defined on the frac…
