Related papers: Interior of distance trees over thin Cantor sets

For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $\Delta^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. Peres and Schlag showed that if the Hausdorff dimension of $E$ is bigger than $\frac{d+2}{2}$ with $d\geq…

In this paper we prove that the set of tuples of edge lengths in $K_1\times K_2$ corresponding to a finite tree has non-empty interior, where $K_1,K_2\subset \mathbb{R}$ are Cantor sets of thickness $\tau(K_1)\cdot \tau(K_2) >1$. Our method…

We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the…

In this paper we show that if a compact set $E \subset \mathbb{R}^d$, $d \geq 3$, has Hausdorff dimension greater than $\frac{(4k-1)}{4k}d+\frac{1}{4}$ when $3 \leq d<\frac{k(k+3)}{(k-1)}$ or $d- \frac{1}{k-1}$ when $\frac{k(k+3)}{(k-1)}…

A theorem of Steinhaus states that if $E\subset \mathbb R^d$ has positive Lebesgue measure, then the difference set $E-E$ contains a neighborhood of $0$. Similarly, if $E$ merely has Hausdorff dimension $\dim_{\mathcal H}(E)>(d+1)/2$, a…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a…

Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset \mathbb{R}^d$. We know from the Mattila-Sj\"olin's theorem if $\dim_H(A)>\frac{d+1}{2d}$, then the distance set $\Delta(E)$ has non-empty interior. In this paper, we show that the…

We show for a compact set $E \subset \mathbb{R}^d$, $d \geq 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty…

In a recent paper, Chan, \L aba, and Pramanik investigated geometric configurations inside thin subsets of the Euclidean set possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside…

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to…

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains…

We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that…

Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\left(A+\Gamma\right)^{\circ}$, when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve and $A\subset…

The classical Cantor's intersection theorem states that in a complete metric space $X$, intersection of every decreasing sequence of nonempty closed bounded subsets, with diameter approaches zero, has exactly one point. In this article, we…

Bennett, Iosevich and Taylor proved that compact subsets of ${\Bbb R}^d$, $d \ge 2$, of Hausdorff dimensions greater than $\frac{d+1}{2}$ contain chains of arbitrary length with gaps in a non-trivial interval. In this paper we generalize…

We prove new results of Mattila-Sj\"olin type, giving lower bounds on Hausdorff dimensions of thin sets $E\subset \Bbb R^d$ ensuring that various $k$-point configuration sets, generated by elements of $E$, have nonempty interior. The…

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 $\alpha$ with…

A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…