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A Cantor set in the plane that is not σ-monotone

Aleš Nekvinda, Ondřej Zindulka (2011)

Fundamenta Mathematicae

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

A class of continua that are not attractors of any IFS

Marcin Kulczycki, Magdalena Nowak (2012)

Open Mathematics

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

Hans Triebel, Heike Winkelvoss (1996)

Studia Mathematica

Let Γ be a closed set in n with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants c 1 > 0 and c 2 > 0 such that c 1 r d µ ( B ( x , r ) ) c 2 r d for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces L p ( Γ ) , 0 < p ≤ ∞, with respect to...

A method for evaluating the fractal dimension in the plane, using coverings with crosses

Claude Tricot (2002)

Fundamenta Mathematicae

Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set E ε , we consider a new method based upon coverings of E ε with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals,...

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ( x ) + 1 / ( d ( x ) d ( x ) ) + . . . + 1 / ( d ( x ) d ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote the Hausdorff...

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