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

We prove that a region in a two-dimensional affine subspace of a normed space $V$ has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits…

Given $E \subset \mathbb{R}^d$, $d \ge 2$, define ${\mathcal D}(E) \equiv {(x-y)/|x-y|: x,y \in E} \subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then…

In this paper we obtain improved dimensional thresholds for dot product sets corresponding to compact subsets of a paraboloid. As a direct application of these estimates, we obtain significant improvements to the best known dimensional…

For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $\delta\in (0,1)$, the $\delta$-neighbourhood of $K$ has Lebesgue measure $\lesssim…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

In this paper it is shown that if $E\subset\mathbb R^{n+1}$ is an $s$-AD regular compact set, with $s\in [n-\frac12,n)$, and $E$ is contained in a hyperplane or, more generally, in an $n$-dimensional $C^1$ manifold, then the Hausdorff…

The Hausdorff dimension of the graphs of the functions in H\"older and Besov spaces (in this case with integrability p \geq 1) on fractal d-sets is studied. Denoting by s \in (0,1] the smoothness parameter, the sharp upper bound…

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

Let $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\mathcal{E}|\gg q^{d-\frac{d-1}{k+1}}$ then $\mathcal{E}$ determines a positive proportion of all…

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…

We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$, such that $$ \left|\sum_{n=1}^N \exp\left(2 \pi i\left(x_1n+\ldots+x_d n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at…

Given $E \subset {\Bbb R}^d$, define the \emph{volume set} of $E$, ${\mathcal V}(E)= \{det(x^1, x^2, ... x^d): x^j \in E\}$. In $\R^3$, we prove that ${\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where…

R\"odl, Ruci\'nski, and Szemer\'edi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and K\"uhn, Osthus, and Townsend extended this result by asymptotically…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

For $d\ge 3$ we first show that the Hausdorff dimension of the set of $A$-divergent on average points in the $(d-1)$-dimensional closed horosphere in the space of $d$-dimensional Euclidean lattices, where $A$ is the group of positive…

We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in…

For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding…