Related papers: A dichotomy for projections of planar sets

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

We characterize when bivariate real analytic functions are "dimension expanding" when applied to a Cartesian product. If $P$ is a bivariate real analytic function that is not locally of the form $P(x,y) = h(a(x) + b(y))$, then whenever $A$…

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset…

The dimension of the visible part of self-affine sets, that satisfy domination and a projection condition, is being studied. The main result is that the assouad dimension of the visible part equals to 1 for all directions outside the set of…

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 show that for any set of reals X there is a subset Y such X and Y have same Lebesgue outer measure and the distance between any two distinct points in Y is irrational.

In this paper we show that given any compact set $E \subset \hat{\mathbb{C}}$, we can always find a conformally removable subset with the same Hausdorff dimension as $E$.

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.…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\,…

Answering an open question raised by Cooper, we show that there exist $\Delta^0_2$ sets $D$ and $E$ such that the singleton degree of $E$ is a minimal cover of the singleton degree of $D$. This shows that the $\Sigma^{0}_{2}$ singleton…

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,…

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$\pi^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this…

In this paper we prove that if f is a planar K-quasiconformal map and 0<t<2, t' = 2t/(2K-Kt+t), then f transforms sets of finite (t')-Hausdorff measure into sets of finite t-Hausdorff measure. We also prove the following more quantitative…

We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a…

In a former paper the first and third authors introduced the notion of direction set for a subset of R^n, and showed that the dimension of the common direction set of two subanalytic subsets, called directional dimension, is preserved by a…

The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms…

Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex…