Related papers: Some Existence Results on Cantor Sets
It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.
Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has…
We show conditions on $k$ such that any number $x$ in the interval $[0, k/2]$ can be represented in the form $x_1^{a_1} x_2^{a_2} + x_3^{a_3} x_4^{a_4} + \cdots + x_{k-1}^{a_{k-1}} x_k^{a_k}$, where the exponents $a_{2i-1}$ and $a_{2i}$ are…
The Cantor distribution is obtained from bitstrings; the Cantor-solus distribution (a new name) admits only strings without adjacent 1 bits. We review moments and order statistics associated with these. The Cantor-multus distribution is…
We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the…
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by…
A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as the unique countable metric space without…
We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers.
We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\to\infty} |n|<na - x> <nb - y> = 0, where <-> denotes the distance from…
In this paper we prove that among pairs $K,\,K' \subset \mathbb{C}$ of conformal dynamically defined Cantor sets with sum of Hausdorff dimensions $HD(K)+HD(K')>2$, there is an open and dense subset of such pairs verifying…
We prove intuitionistic versions of the classical theorems saying that all countable closed subsets of $[-\pi,\pi]$ and even all countable subsets of $[-\pi,\pi]$ are sets of uniqueness.
We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-\Delta w_1=2e^{2w_1}-e^{w_2}+2\pi\sum_{\ell=1}^m\beta_{1,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\…
Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets…
In this article we give necessary and sufficient conditions that a complex number must satisfy to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and…
We show that the closure of any self-entwined ray in the plane must contain a Cantor set of mutually disjoint continua. This is false in dimension three.
We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to…
We prove that any two countable, compact, subsets of $\mathbb{S}^n, n\geq 2$ that are homeomorphic also have homeomorphic complements. Thus any wild subspace like the classical construction of Antoine must contain a Cantor set.
We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supporting this heuristic.
We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.