Related papers: On necklaces inside thin subsets of ${\Bbb R}^d$

Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$…

It is known that if a compact set $E$ in $\mathbb{R}^d$ has Hausdorff dimension greater than $(d+1)/2$, then its $n$-chain distance set $$\Delta^n(E) = \left\{\left(\left|x^1-x^2\right|,\cdots, \left|x^{n}- x^{n+1}\right|\right)\in…

Let an $R$-body be the complement of the union of open balls of radius $R$ in $\mathbb{E}^d$. The $R$-hulloid of a closed not empty set $A$, the minimal $R$-body containing $A$, is investigated; if $A$ is the set of the vertices of a…

We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least…

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 us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…

For any composant $E \subset \mathbb H^*$ and corresponding near-coherence class $\mathscr E \subset \omega^*$ we prove the following are equivalent : (1) $E$ properly contains a dense semicontinuum. (2) Each countable subset of $E$ is…

Let $E \subseteq R^n$ be a closed set of Hausdorff dimension $\alpha$. For $m \geq n$, let $\{B_1,\ldots,B_k\}$ be $n \times (m-n)$ matrices. We prove that if the system of matrices $B_j$ is non-degenerate in a suitable sense, $\alpha$ is…

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - \epsilon_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \}…

For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property that it is…

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an…

We study the topdrop map, a mapping on permutations in $S_n$ related to card shuffling. We show this map is bijective and study its orbit structure. We introduce the notion of the topdrop-necklace as a way of classifying the orbits of the…

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by…

We prove that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^2$ is greater than 7/4, then the set of {\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing…

The classical Erd\H{o}s--Ko--Rado (EKR) theorem characterizes the maximum size of intersecting families of $r$-element subsets of an $n$-element set. We study EKR-type questions for independent $r$-sets in \emph{pendant} graph…

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…

The Falconer distinct distance problem asks for a compact set $E\subset\mathbb{R}^d$ how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of its distance set is positive. In this paper we consider the analogous…

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…

In hypergraph theory, determining a characterization of the degree sequence $d=(d_1,d_2,\ldots,d_n)$ where $d_1\ge d_2\ge\ldots,d_n$ are positive integers, of an $h$-uniform simple hypergraph $\cal H$, and deciding the complexity status of…

Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.