2023
DOI: 10.54330/afm.126014
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When the algebraic difference of two central Cantor sets is an interval?

Abstract: Let \(C(a ),C(b)\subset \lbrack 0,1]\) be the central Cantor sets generated by sequences \(a,b \in \left( 0,1\right)^{\mathbb{N}}\). The first main result of the paper gives a necessary and a sufficient condition for sequences \(a\) and \(b\) which inform when \(C(a )-C(b)\) is equal to \([-1,1]\) or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets, the algebraic difference of which is an interval, may be … Show more

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Cited by 2 publications
(4 citation statements)
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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%
“…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%
“…Very similar intervals were defined firstly in [15]. They were also used in [8] and [13]. Put J := I − I = [−1, 1].…”
Section: Preliminariesmentioning
confidence: 99%
“…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]. Later this result was generalized several times (see [11], [3], [8], [13], [2]). Also, there are results on the possible form of algebraic difference of Cantor sets.…”
Section: Introductionmentioning
confidence: 99%
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