Related papers: A generalised isodiametric problem

A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer…

We give a simple and short proof of the classical Lebesgue decomposition theorem of measures via the Riesz orthogonal decomposition theorem of Hilbert spaces. The tools we employ are elementary Hilbert space techniques.

We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper…

A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates…

A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erd\H{o}s and…

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for…

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the…

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to…

We show that if $\beta>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^{\beta}+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of…

A result of P. Tukia from 1989 says that Lebesgue measure on $\mathbb{R}$ has conformal dimension zero: for every $\epsilon > 0$, there is a Borel set $G \subset \mathbb{R}$ of full Lebesgue measure, and a quasisymmetric homeomorphism $f…

We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the…

We consider a special one-parameter family of d-dimensional random, homogeneous self-similar iterated function systems (IFSs) satisfying the finite type condition. The object of our study is the positivity of Lebesgue measure and the…

Let $K\subset \mathbb R$ be a regular compact set and let $g(z)=g_{\overline{\mathbb C}\setminus K}(z,\infty)$ be the Green function for $\overline{\mathbb C}\setminus K$ with pole at infinity. For $\delta>0$, define $$ G(\delta):=\max\{…

P\'al's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $\tfrac \pi 2$. If the width…

We prove that on a $d$-dimensional Riemannian manifold, the distance set of a Borel set $E$ has a positive Lebesgue measure if $$\dim_{\mathcal{H}}(E)>\frac d2+\frac14+\frac{1-(-1)^d}{8d}.$$

We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the…

The Erd\H{o}s similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences.…

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the…