Slicing the Sierpiński gasket

Bárány, Balázs; Ferguson, Andrew; Simon, Károly

doi:10.1088/0951-7715/25/6/1753

Cited by 18 publications

(21 citation statements)

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“…for all lines ℓ with irrational slope. The intersections of these carpets with lines of rational slope was investigated in several papers; see [3,4] and references there. In particular, in those two papers it is shown that for the gasket and many other carpets S, there are many lines with a given rational slope that intersect S in a set of dimension > dim H (S) − 1.…”

Section: Some Applicationsmentioning

confidence: 99%

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

2019

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We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's longstanding conjecture on the dimension of the intersections of ×p and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an L q density for all finite q, outside of a zero-dimensional set of exceptions.The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to L q norms, and likewise relies on an inverse theorem for the decay of L q norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

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Section: Some Applicationsmentioning

confidence: 99%

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

2019

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“…Proof. The part (1) is clear, we only need to prove (2). By definition, π t µ is a measure supported on some slice of K with the form K ∩ ℓ β −t ,z for some z ∈ K. It is clear that, for all n ≥ 1 and each element A of A t n , the support of π t µ intersects the boundary of A in at most two points.…”

Section: 1mentioning

confidence: 99%

“…For A p , B q as in Conjecture 1.1, the set A p × B q is such an example, for which it is clear that certain lines parallel to the axes are exceptional for the slice result, and Conjecture 1.1 predicts that these lines are the only exceptions.There is a rich literature about generic slices of various fractal sets, see e.g. [2,7,8,20,26,28,29,37]. However, very little is known about specific slices, and there were few partial results concerning Conjecture 1.1 before the present paper.…”

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confidence: 95%

A proof of Furstenberg's conjecture on the intersections of $\times p$- and $\times q$-invariant sets

2019

Ann. of Math. (2)

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We prove the following conjecture of Furstenberg (1969): if A, B ⊂ [0, 1] are closed and invariant under ×p mod 1 and ×q mod 1, respectively, and if log p/ log q / ∈ Q, then for all real numbers u and v,We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on R. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions. 1 or combinatorial reasons. For A p , B q as in Conjecture 1.1, the set A p × B q is such an example, for which it is clear that certain lines parallel to the axes are exceptional for the slice result, and Conjecture 1.1 predicts that these lines are the only exceptions.There is a rich literature about generic slices of various fractal sets, see e.g. [2,7,8,20,26,28,29,37]. However, very little is known about specific slices, and there were few partial results concerning Conjecture 1.1 before the present paper. The first and perhaps also the best one is due to Furstenberg [16, Theorem 4]. His result states that under the assumption of the conjecture, if dimFrom the last assertion, it is not hard to deduce that in this case, we must have dim H A p + dim H B q > 1/2 (see [22, Theorem 7.9] for the deduction). Thus, under the assumption dim H A p + dim H B q ≤ 1/2, Furstenberg's result confirms Conjecture 1.1. We will return back to [16, Theorem 4] in Subsection 4.2. We would like to mention that the technique (namely, CP-process) Furstenberg introduced and used in [16] is also important for the present work, it will be one of the main ingredients for our proof of Conjecture 1.1.Recently, Feng, Huang and Rao [13] studied affine embeddings between incommensurable self-similar sets and, as a consequence, they showed that if p ≁ q, then for T p -invariant self-similar set E and T q -invariant self-similar set F , there exists a (non-effective) positive constant δ depending on E and F such that the Hausdorff dimension of the intersection Later, Feng [12] obtained some effective versions of the results of [13], but these effective versions are still far from sufficient for proving Conjecture 1.1. Feng [12] also constructed, for any s, t ∈ (0, 1) and ǫ > 0, a T p -invariant set A of dimension s and a T q -invariant set B of dimension t which verify Conjecture 1.1 with a loss of ǫ.Finally, we note that the slice problem may be considered as "dual" to the projection problem for fractal sets. In that direction, there is a dual version of Conjecture 1.1, also due to Furstenberg and recently settled by Hochman and Shmerkin [23] (some special cases by Peres and Shmerkin [31]), which asserts that under the assumptions of Conjecture 1.1, for each orthogonal projection P θ from R 2 to R with direction θ not parallel to the axes, we haveRecently, there has been considerable interest in the study of projections of dynamically defined Cantor sets, see for instance the survey paper of Shmerkin [35] and the references therein for more details.1.2. Statem...

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“…In general, the relation between the Hausdorff dimensions of a self-affine set E ⊂ R n and of its intersection with a hyperplane S has been studied by many authors (Bárány, Ferguson and Simon [1], Manning and Simon [10], for example). In a recent study by Kempton [9], it is proved that if E is a self-similar set with dim H E = s such that the similarities do not include rotations and that the orthogonal projection of the s-dimensional Hausdorff measure on E to a line θ is absolutely continuous with a bounded density, then H s−1 (E ∩ S) > 0 and hence, dim H (E ∩ S) ≥ s − 1 holds for almost all S perpendicular to θ.…”

Section: Definitionmentioning

confidence: 99%

“…Images of the mappings ϕ I,1 × ϕ J,1 (left) and ϕ I,1 × ϕ J,−1 (right). where |I| denotes the length b − a of the interval I = [a, b].…”

mentioning

confidence: 99%

Hausdorff dimension of the level sets of self-affine functions

Peng

Kamae

2015

Journal of Mathematical Analysis and Applications

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Given a partition of [0, 1] by closed intervals I 1 ∪ · · · ∪ I k = [0, 1] with k ≥ 2. Given 0 < α < 1 and closed intervalswhere for an interval I = [a, b] ⊂ [0, 1] and τ ∈ {−1, 1}, ϕ I,τ : [0, 1] → I is the linear map such that ϕ I,1 (0) = a, ϕ I,1 (1) = b and ϕ I,−1 (0) = b, ϕ I,−1 (1) = a. Such Ω is a graph of a Borel function f Ω almost surely and is called a self-affine set of α-function type. We obtain the Hausdorff dimension of the level set Ω y = {x; (x, y) ∈ Ω} in the case that λ • f −1 Ω has a bounded density with respect to λ, where λ is the Lebesgue measure on [0, 1].

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Slicing the Sierpiński gasket

Cited by 18 publications

References 20 publications

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

A proof of Furstenberg's conjecture on the intersections of $\times p$- and $\times q$-invariant sets

Hausdorff dimension of the level sets of self-affine functions

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