The Lebesgue measure of some M-Cantorval

Banakiewicz, Michał

doi:10.1016/j.jmaa.2018.10.070

Cited by 3 publications

(4 citation statements)

References 9 publications

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“…In particular, any Cantorval is homeomorphic to the Guthrie-Nymann Cantorval given by E(z n ), with z 2n = 2/4 n and z 2n−1 = 3/4 n , the one originally considered in (iii) of the above Theorem. Cantorvals also appear as attractors associated to some iterated function systems [2], and an analogous result to Theorem 1.1 holds to describe the topology of the algebraic difference of certain Cantor sets [1,10,12].…”

Section: Introductionmentioning

confidence: 86%

The Achievement Set of Generalized Multigeometric Sequences

Karvatskyi,

Murillo,

Viruel

2024

Results Math

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We study the topology of all possible subsums of the generalized multigeometric series$$k_1f(x)+k_2f(x)+\dots +k_mf(x)+\dots + k_1f(x^n)+\dots +k_mf(x^n)+\dots ,$$ k 1 f ( x ) + k 2 f ( x ) + ⋯ + k m f ( x ) + ⋯ + k 1 f ( x n ) + ⋯ + k m f ( x n ) + ⋯ , where $$k_1, k_2, \dots , k_m$$ k 1 , k 2 , ⋯ , k m are fixed positive real numbers and f runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.

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Section: Introductionmentioning

confidence: 86%

The Achievement Set of Generalized Multigeometric Sequences

Karvatskyi,

Murillo,

Viruel

2024

Results Math

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“…we obtain (7). If λ n+1 ≥ 1 3 , then by (6), we have J s = J sˆ0 ∪ J sˆ1 ∪ J sˆ2 . Consequently, J s = t∈{0,1,2} n+1 ,s≺t J t , which implies (8).…”

Section: Introductionmentioning

confidence: 95%

“…Recently, there have appeared a lot of interesting papers that explore properties of achievement sets, i.a. [5], [7], [27], [9], and [6].…”

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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“…Recently, there have appeared a lot of interesting papers that explore properties of achievement sets, i.e. [5][6][7]9,23,29]. Recall the notion of a Cantorval.…”

Section: Introductionmentioning

confidence: 99%

Conditions for the Difference Set of a Central Cantor Set to be a Cantorval

Nowakowski

Filipczak

2023

Results Math

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Let $$C(\lambda )\subset [0,1]$$ C ( λ ) ⊂ [ 0 , 1 ] denote the central Cantor set generated by a sequence $$\lambda =(\lambda _{n})\in \left( 0,\frac{1}{2} \right) ^{\mathbb {N}}$$ λ = ( λ n ) ∈ 0 , 1 2 N . By the known trichotomy, the difference set $$ C(\lambda )-C(\lambda )$$ C ( λ ) - C ( λ ) of $$C(\lambda )$$ C ( λ ) is one of three possible sets: a finite union of closed intervals, a Cantor set, or a Cantorval. Our main result describes effective conditions for $$(\lambda _{n})$$ ( λ n ) which guarantee that $$C(\lambda )-C(\lambda )$$ 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 $$C(\lambda )-C(\lambda )$$ 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 Lebesgue measure of some M-Cantorval

Cited by 3 publications

References 9 publications

The Achievement Set of Generalized Multigeometric Sequences

The Achievement Set of Generalized Multigeometric Sequences

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

Conditions for the Difference Set of a Central Cantor Set to be a Cantorval

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