Multiplication on uniform λ-Cantor sets

Gu, Jiangwen; Jiang, Kan; Xi, Lifeng; Zhao, Bing

doi:10.5186/aasfm.2021.4643

Cited by 7 publications

(1 citation statement)

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“…One of research areas concerning Cantor sets is related to examining how does the algebraic difference (or sum, or product) looks like (e.g. [17], [18], [10], [1], [9]). One of the most known theorems in this topic is the classical result of Steinhaus ([16]), which states that the difference set of the classical Cantor set is [−1 , 1].…”

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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