Related papers: Improvement on $2$-chains inside thin subsets of E…

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…

Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets $\{|x-y| : x,y\in E\}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive…

We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying $\mathscr{H}^{d}_{\infty}(E\cap…

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…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

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…

We prove a ''dimension expansion'' version of the Elekes-R\'onyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is…

Let $\mathbb{F}_q$ be a finite field of order $q$ and $E$ be a set in $\mathbb{F}_q^d$. The distance set of $E$ is defined by $\Delta(E):=\{\lVert x-y \rVert :x,y\in E\}$, where $\lVert \alpha \rVert=\alpha_1^2+\dots+\alpha_d^2$. Iosevich,…

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 investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has…

In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\textbf{d}}=(d_1,d_2, \dots, d_{\ell})\in \mathbb{N}^{\ell}$ with $d_1+d_2+\dots+d_{\ell}=d$ and $E \subseteq \mathbb{R}^d$,…

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…

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…

Let $E\subset R^d$ with $H^n(E)<\infty$, where H^n stands for the $n$-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit $$\lim_{\ve\to0}\int_{y\in E:|x-y|>\ve} \frac{x-y}{|x-y|^{n+1}}…

We characterize the gauge profile of $\mathcal{D}_s$, the set of reals with effective dimension $s$, and $\mathcal{D}_{\leq s}$, the set of reals with effective dimension $\leq s$. Let $W(s)$ be the set of reals that are $s$-well…

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…

We study the distance set problem for pairs of compact sets $A, B\subset \mathbb{R}^n$, $n\geq 2$. We show that if $B$ is contained in a hyperplane and \begin{align*} \dim_{H} A+\dim_{H} B>n, \end{align*} then the distance set $…

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

For Borel subsets $\Theta\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} \Theta(E):=\bigcup_{(g,z)\in \Theta}(gE+z). \end{align*} In this paper, we investigate the…

We prove that, for every norm on $\mathbb{R}^d$ and every $E \subseteq \mathbb{R}^d$, the Hausdorff dimension of the distance set of $E$ with respect to that norm is at least $\dim_{\mathrm{H}} E - (d-1)$. An explicit construction follows,…