In this survey we collect and discuss some recent results on the so called "Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval poitning in that direction of some prescribed dimension. This problem is closely related to the "Kakeya needle problem". In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of "logarithmic gaps" and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.
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…
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 dim…
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…
In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.
We consider dimensional properties of limit sets of Moran constructions satisfying the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the weak separation condition and certain sub-self-affine sets. In addition to dimension results for the limit set, we manage to express the Assouad dimension of any closed subset of a self-conformal set by means of the Hausdorff dimension. As an interesting consequence of this, we show that a Furstenberg homogeneous self-similar set in the real line satisfies the weak separation condition. We also exhibit …
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.
For $α$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $α$ or $F_α$-set if for each direction $e$ in the unit circle 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 greater or equal than $α$. In this paper we show that if $α> 0$, there exists a set $E\in F_α$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2α}\log^{-θ}(\frac{1}{x})$, $θ>\frac{1+3α}{2}$, which improves on the the previously known bound, that $H^β(E) = 0$ for $β>1/2+3/2α$. Further, by refining the argument in a subtle way, we are able to obtai…
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 use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point (x, ax+b), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.
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 …
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine …
A classical theorem due to Mattila (see \cite{Mat84}; see also \cite{M95}, Chapter 13) says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively, with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{\mathcal H}(A \times B)=s_A+s_B\ge d$, then $$ dim_{\mathcal H}(A \cap (z+B)) \leq s_A+s_B-d$$ for almost every $z \in {\Bbb R}^d$, in the sense of Lebesgue measure. In this paper, we replace the Hausdorff dimension on the left hand side of the first inequality above by the upper Minkowski dimension, replace the Lebesgue measure of the set of translates by a Hausdorff mea…
On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；National Taiwan University（台湾大学）；VNU University of Science, Hanoi, Vietnam（越南河内国立大学科学大学）
In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying c…
We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $θ$-spectrum are other special examples of these intermediate dimensions. These dimensions are localized, like Assouad dimensions, but vary in the depth of scale which is considered, thus they provide very refined geometric information. We investigate the relationship between these and the familiar dimensions. We construct a Cantor set with a non-trivial interval of dimensions, the endpoints of this interval being given by the quasi-Assouad and Assouad di…
We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directio…
