On the size of the algebraic difference of two random Cantor sets

Dekking, Michel; Simon, Károly

doi:10.1002/rsa.20178

Cited by 15 publications

(43 citation statements)

References 9 publications

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“…The proof is similar to the proof of Lemma 1 in [DS08], but it shows where and how the JSC emerges for general survival distributions. .…”

Section: Joint Growth Of -Pairsmentioning

confidence: 77%

“…A simple geometric analysis shows that F = [−1/2, 1/2], and hence F must contain this interval. (An algebraic alternative is to use Theorem 2 of [DS08]: the collection of reduced matrices ofμ is {T 3 , T 9 , T 12 }, and since this set is closed under (mutual) multiplications, this theorem tells us thatF will contain an interval.) See [DD10] for a weaker condition than the joint survival condition under which Theorem 4.1 still holds.…”

Section: The Basic Results With Joint Survival Distributionsmentioning

confidence: 99%

“…The tool that is used in the proof of this result is that of higher order Cantor sets, which has been introduced in [DS08]. The same tool will enable us in Section 7 to obtain a general classification result in terms of the lower spectral radius of a certain set of matrices.…”

Section: Theorem 11 ([Ds08]mentioning

confidence: 99%

“…Lemma 5.1 (Extension of Lemma 1 in [DS08]). If γ > 1, and the joint survival distribution(s) satisfy the joint survival condition, then for all n ≥ 0,…”

Section: Joint Growth Of -Pairsmentioning

confidence: 99%

“…The (rather embarrassing) reason is that the equality P(A c n+1 | A r ∩ · · · ∩ A n ) = P(A c n+1 | A n ) on the bottom of page 10 in [DS08] is wrong in general. All one needs is that the equality sign can be replaced by a ≤ sign, but since proving this seems to be rather involved (although intuitively obvious) we chose to redefine A n as in (20), since this conveniently leads to a sequence of decreasing sets with intersection the set in the statement of Lemma 5.4.…”

Section: Exponential Growthmentioning

confidence: 99%

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Differences of random Cantor sets and lower spectral radii

Dekking¹,

Kuijvenhoven²

2011

J. Eur. Math. Soc.

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Abstract. We investigate the question under which conditions the algebraic difference between two independent random Cantor sets C 1 and C 2 almost surely contains an interval, and when not. The natural condition is whether the sum d 1 + d 2 of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that generically it should be true that d 1 + d 2 > 1 should imply that C 1 − C 2 contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities (p 0 , p 1 ) the interior of the region where the Palis conjecture does not hold is given by those p 0 , p 1 which satisfy p 0 + p 1 > √ 2 and p 0 p 1 (1 + p 2 0 + p 2 1 ) < 1. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of 2 × 2 matrices.

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“…The proof is similar to the proof of Lemma 1 in [DS08], but it shows where and how the JSC emerges for general survival distributions. .…”

Section: Joint Growth Of -Pairsmentioning

confidence: 77%

Section: The Basic Results With Joint Survival Distributionsmentioning

confidence: 99%

Section: Theorem 11 ([Ds08]mentioning

confidence: 99%

“…Lemma 5.1 (Extension of Lemma 1 in [DS08]). If γ > 1, and the joint survival distribution(s) satisfy the joint survival condition, then for all n ≥ 0,…”

Section: Joint Growth Of -Pairsmentioning

confidence: 99%

Section: Exponential Growthmentioning

confidence: 99%

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Differences of random Cantor sets and lower spectral radii

Dekking¹,

Kuijvenhoven²

2011

J. Eur. Math. Soc.

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Random Cantor Sets and Their Projections

Dekking

2009

Fractal Geometry and Stochastics IV

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The Geometry of Fractal Percolation

Rams

Simon

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:• the statements work for all directions, not almost all,• the statements are true for more general projections, for example radial projections onto a circle, • in the case dim H > 1, each projection has not only positive Lebesgue measure but also has nonempty interior. introductionTo model turbulence, Mandelbrot [13, 14] introduced a statistically self-similar family of random Cantor sets. Since that time this family has got at least three names in the literature: fractal percolation, Mandelbrot percolation and canonical curdling, among which we will use the first one. In 1996 Lincoln Chayes [3] published an excellent survey giving an account about the most important results known in that time. His survey focused on the percolation related properties while we place emphasis on the geometric measure theoretical properties (projections and slices) of fractal percolation sets. About the projections of a general Borel set the celebrated Marstrand Theorem gives the following information:2000 Mathematics Subject Classification. Primary 28A80 Secondary 60J80, 60J85Key words and phrases. Random fractals, Hausdorff dimension, processes in random environment.

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On the size of the algebraic difference of two random Cantor sets

Cited by 15 publications

References 9 publications

Differences of random Cantor sets and lower spectral radii

Differences of random Cantor sets and lower spectral radii

Random Cantor Sets and Their Projections

The Geometry of Fractal Percolation

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