Related papers: An elementary approach to simplexes in thin subset…

Kupavskii proved a codegree version of the Erd\H{o}s--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most…

We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\cal{H}^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of…

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parametrizing matrix. This provides evidence towards the conjecture that all affine subspaces…

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$…

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.…

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 <…

We prove partial regularity for minimizers to elasticity type energies in the nonlinear framework {with $p$-growth, $p>1$,} in dimension $n\geq 3$. It is an open problem in such a setting either to establish full regularity or to provide…

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…

We prove a strengthening of the trickle down theorem for partite complexes. Given a $(d+1)$-partite $d$-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are $\frac{1-\delta}{d}$-(one-sided)…

We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…

Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Gr\"unbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$.…

It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the…

In this paper we prove that the singular set of connected minimizers of the planar Griffith functional has Hausdorff dimension strictly less then one, together with the higher integrability of the symetrized gradient.

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…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

Based on a result of Makeev, in 2012 Blagojevi\'c and Karasev proposed the following problem: given any positive integers $m$ and $1\leq \ell\leq k$, find the minimum dimension $d=\Delta(m;\ell/k)$ such that for any $m$ mass distributions…

In this paper we prove that if $\phi:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

Let $M$ be a compact $d$-dimensional Riemannian manifold without a boundary. Given $E \subset M$, let $\Delta_{\rho}(E)=\{\rho(x,y): x,y \in E \}$, where $\rho$ is the Riemannian metric on $M$. Let $\Delta_{\rho}^x$ denote the pinned…