Related papers: Diophantine sets in general are Cantor sets

A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the…

Every element $u$ of $[0,1]$ can be written in the form $u=x^2y$, where $x,y$ are elements of the Cantor set $C$. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and…

A finitely generated subgroup {\Gamma} of a real Lie group G is said to be Diophantine if there is \beta > 0 such that non-trivial elements in the word ball B_\Gamma(n) centered at the identity never approach the identity of G closer than…

Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely…

We introduce a class of self-similar sets which we call {\em twofold Cantor sets} $K_{pq}$ in $\mathbb R$ which are totally disconnected, do not have weak separation property and at the same time have isomorphic self-similar structures.

Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de…

We consider the self-similar structure of the class of generalized Cantor sets $$\Gamma_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_n\beta^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<\beta<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of…

We continue to study topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology tau, which was started in the paper (S. Bezuglyi, A.H. Dooley, and J. Kwiatkowski, Topologies on…

We investigate some Diophantine approximation constants related to the simultaneous approximation of $(\zeta,\zeta^{2},\ldots,\zeta^{k})$ for Liouville numbers $\zeta$. For a certain class of Liouville numbers including the famous…

A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. In this language a Diophantine figure is a complete Diophantine graph. Due…

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there…

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first…

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…

Let $\Gamma$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of…

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger…

For any $\alpha\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\| \widehat{f d\mu} \|_{p} \leq C_p \| f \|_{L^2(\mu)}$ for…

The approximation constant $\lambda_{k}(\zeta)$ is defined as the supremum of real $\eta$ such that $\Vert \zeta^{j}x\Vert\leq x^{-\eta}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance…