Related papers: Point-to-set Principle and Constructive Dimension …

Given a positive integer $M$ and a real number $q>1$, we consider the univoque set $\mathcal{U}_q$ of reals which have a unique $q$-expansion over the alphabet $\set{0,1,\cdots,M}$. In this paper we show that for any $x\in\mathcal{U}_q$ and…

In this paper, we answer a question of Nussbaum, Priyadarshi, and Lunel [Positive operators and Hausdorff dimension of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066.]. We also show that the Hausdorff dimension and box…

Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations under the largeness of sets, which are sometimes specified via the ball condition and Fourier decay. Recently,…

The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known…

We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can…

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general…

This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained…

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension…

Extensions of Huber's Theorem to higher dimensions with $L^\frac{n}{2}$ bounded scalar curvature have been extensively studied over the years. In this paper, we delve into the properties of conformal metrics on a punctured ball with…

In this paper we study a quantitative notion of exactness within Diophantine approximation. Given $\Psi:(0,\infty)\to (0,\infty)$ and $\omega:(0,\infty)\to (0,1)$ satisfying $\lim_{q\to\infty}\omega(q)=0$, we study the set of points, which…

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial…

Z. Wen and J. Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their fundamental intervals and the gaps between them. In this…

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain…

We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a…

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…

In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff…

We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation…

We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U|…

We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of…