Related papers: Projection inequalities for antichains

A subset $A$ of $\mathbb{Z}^n$ is called a weak antichain if it does not contain two elements $x$ and $y$ satisfying $x_i<y_i$ for all $i$. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain $A$, the sum of the sizes of…

An \emph{$n$-cube antichain} is a subset of the unit $n$-cube $[0,1]^n$ that does not contain two elements $\mathbf{x}=(x_1, x_2,\ldots, x_n)$ and $\mathbf{y}=(y_1, y_2,\ldots, y_n)$ satisfying $x_i\le y_i$ for all $i\in \{1,\ldots,n\}$.…

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$.…

It is well-known that an antichain in the poset $[0,1]^n$ must have measure zero. Engel, Mitsis, Pelekis and Reiher showed that in fact it must have $(n-1)$-dimensional Hausdorff measure at most $n$, and they conjectured that this bound can…

A family of sets $A$ is said to be an antichain if $x\not\subset y$ for all distinct $x,y\in A$, and it is said to be a distance-$r$ code if every pair of distinct elements of $A$ has Hamming distance at least $r$. Here, we prove that if…

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$.…

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the…

We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally…

It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both…

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in…

A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichain if $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor…

We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…

The Besicovitch projection theorem states that if a subset $E$ of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every…

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$…

In a previous work we proved that if a finite Borel measure $\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is…

The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every…

Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erd\H{o}s, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \[ A=\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1]\}, \] where…

This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every…

For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted'…

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to…