On the arithmetic sums of Cantor sets

Eroglu, Kemal Ilgar

doi:10.1088/0951-7715/20/5/005

Cited by 13 publications

(11 citation statements)

References 14 publications

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“…In [17] (as part of a more general framework) the dimension of the set of exceptions in (2) was estimated, but whether the set of exceptions (for K = C b ) is countable, was unknown until now. Eroǧlu [3] proved that if dim(C a ) + dim(C b ) ≤ 1, then the Hausdorff measure of the sum set C a + C b in the dimension dim(C a ) + dim(C b ) is zero.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We can assume without loss of generality that f i (B) ⊂ B for all i, where B is the unit ball. Fix some small ε > 0 and some large m. We can apply Proposition 7 to the family Q = {f u (B) : |u| = m}, with ρ = ζ m , γ = dim(E), and some A, A 1 , A 2 which depend only on E. In this way we obtain a set of "good" angles J verifying properties (1)- (3).…”

Section: Consider the Eventsmentioning

confidence: 99%

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Resonance between Cantor sets

Peres

Shmerkin

2009

Ergod. Th. Dynam. Sys.

101

129

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Let Ca be the central Cantor set obtained by removing a central interval of length 1 − 2a from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if log b/ log a is irrational, thenwhere dim is Hausdorff dimension. More generally, given two self-similar sets K, K ′ in R and a scaling parameter s > 0, if the dimension of the arithmetic sum K + sK ′ is strictly smaller than dim(K) + dim(K ′ ) ≤ 1 ("geometric resonance"), then there exists r < 1 such that all contraction ratios of the similitudes defining K and K ′ are powers of r ("algebraic resonance"). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Consider the Eventsmentioning

confidence: 99%

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Resonance between Cantor sets

Peres

Shmerkin

2009

Ergod. Th. Dynam. Sys.

101

129

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“…Also, it is obvious that if HD(C α ) + HD(C β ) < 1 and log α log β ∈ Q, then (7) is always valid [11]. Moreover, H HD(Cα)+HD(C β ) (C α + C β ) = 0 ( [5]).…”

Section: Regardless Of the Indexes We Havementioning

confidence: 99%

On the arithmetic difference of middle Cantor sets

Pourbarat¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Suppose that C is the space of all middle Cantor sets. We characterize all triples (α, β, λ) ∈ C × C × R * that satisfy C α − λC β = [−λ, 1]. Also all triples (that are dense in C × C × R * ) has been determined such that C α − λC β forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets P in a way that for each (C α , C β ) ∈ P, there exists a dense subfield F ⊂ R such that for each µ ∈ F , the set C α − µC β contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions f, g and the pair (C α , C β ) is provided which f (C α ) − g(C β ) contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one. At the end, special middle Cantor sets C α and C β are introduced. Then the iterated function system corresponding to the attractor C α − 2α β C β is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that √ C α -C β contains at least one interval.

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“…Besides µ a * µ b , other possible natural measures on C a + C b are the restrictions of Hausdorff and packing measures in the appropriate dimension. However, Eroǧlu [2] proved that H da+d b (C a + C b ) = 0, for all a, b such that d a + d b ≤ 1. We do not know whether C a + C b has positive d a + d b -dimensional packing measure when log b/ log a is irrational (it is easy to see that it is finite).…”

Section: Remarks and Open Questionsmentioning

confidence: 99%

Convolutions of cantor measures without resonance

Nazarov¹,

Peres

Shmerkin

2012

Isr. J. Math.

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Abstract. Denote by µ a the distribution of the random sum (1 − a) ∞ j=0 ω j a j , where P(ω j = 0) = P(ω j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ a is supported on C a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 − 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ a * (µ b • S −1 λ ), where S λ (x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ = 0, thenwhere D denotes any of correlation, Hausdorff or packing dimension of a measure.We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ 1/4 * (µ 1/3 •S −1 λ ) is a singular measure, although dim H (C

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On the arithmetic sums of Cantor sets

Cited by 13 publications

References 14 publications

Resonance between Cantor sets

Resonance between Cantor sets

On the arithmetic difference of middle Cantor sets

Convolutions of cantor measures without resonance

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