Exact packing measure of central Cantor sets in the line

García, Ignacio; Zuberman, Leandro

doi:10.1016/j.jmaa.2011.08.044

Cited by 12 publications

(16 citation statements)

References 13 publications

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“…This section is devoted to proving Theorem 1. One of the difficulties one needs to overcome to show the rate of convergence (4) is to obtain a comparison between the measures µ and µ k of a given ball (see (8) and (15)). Note that to obtain a bound for |P s (E) −M k | we need to compare the densities h(x, d) given in (12) with those given in (19).…”

Section: 2mentioning

confidence: 99%

“…We denote by µ the natural probability measure, or normalized Hausdorff measure, defined on the ring of cylinder sets by (8) µ…”

Section: Notationalmentioning

confidence: 99%

“…The similarity ratio of f i ∈ Ψ is denoted by r i ∈ (0, 1), and we write Both algorithms are based on the self-similar tiling principle stated in [17] for self-similar sets satisfying the OSC. In [17] it was shown that if B is any closed subset (or tile) of a self-similar set E such that µ(B) > 0, where µ is the invariant measure (see (8)), then E can be tiled, without any loss of µ-measure, by a countable collection of tiles that are images of B under similitudes. Recall from [9] that, for selfsimilar sets satisfying the OSC, the measure µ is a multiple of any scaling measure, and in particular of the packing, Hausdorff, or centered Hausdorff measure.…”

Section: Introductionmentioning

confidence: 99%

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Rate of convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets

Llorente

Mera

Morán

2017

Chaos, Solitons & Fractals

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Abstract. In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing measures of a totally disconnected self-similar set. We evaluate these rates empirically through the numerical analysis of three standard classes of self-similar sets, namely, the families of Cantor type sets in the real line and the plane and the class of Sierpinski gaskets. For these three classes and for small contraction ratios, sharp bounds for the exact values of the corresponding measures are obtained and it is shown how these bounds automatically yield estimates of the corresponding measures, accurate in some cases to as many as 14 decimal places. In particular, the algorithms accurately recover the exact values of the measures in all cases in which these values are known by geometrical arguments. Positive results, which confirm some conjectural values given in [13] and [14] for the measures, are also obtained for an intermediate range of larger contraction ratios. We give an argument showing that, for this range of contraction ratios, the problem is inherently computational in the sense that any theoretical proof, such as those mentioned above, might be impossible, so that in these cases, our method is the only available approach. For contraction ratios close to those of the connected case our computational method becomes intractably time consuming, so the computation of the exact values of the packing and centered Hausdorff measures in the general case, with the open set condition, remains a challenging problem.

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Section: 2mentioning

confidence: 99%

“…We denote by µ the natural probability measure, or normalized Hausdorff measure, defined on the ring of cylinder sets by (8) µ…”

Section: Notationalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rate of convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets

Llorente

Mera

Morán

2017

Chaos, Solitons & Fractals

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“…Finally, let us mention the applications in spectral theory [11], [30]. Important results describe geometrical properties of Cantor sets via Hausdorff and packing measures and the respective fractal dimensions, and multifractal spectrum [3], [10], [15], [17]. Also, several authors conducted extensive studies on the arithmetic sums [1], [22], [31] and products [29] of two Cantor sets, and their intersections [18], [20].…”

Section: Introductionmentioning

confidence: 99%

Conditions for the difference set of a central Cantor set to be a Cantorval

Nowakowski¹,

Filipczak²

2020

Preprint

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Let C(λ) ⊂ [0, 1] denote the central Cantor set generated by a sequence λ = (λn) ∈ 0, 1 2 N . By the known trichotomy, the difference set C(λ) − C(λ) of C(λ) is one of three possible sets: a finite union of closed intervals, a Cantor set, and a Cantorval. Our main result describes effective conditions for (λn) which guarantee that C(λ) − C(λ) is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval C(λ) − C(λ) is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.

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“…The notion of a central (or symmetric) Cantor set in the real line is well known and widely explored by several authors (see e.g. [8]). Its construction mimics that of the classical ternary Cantor subset of [0, 1].…”

mentioning

confidence: 99%