Differentiability of fractal curves

Bandt, Christoph; Kravchenko, A. S.

doi:10.1088/0951-7715/24/10/003

Cited by 17 publications

(33 citation statements)

References 12 publications

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“…See Section 13.5. (7) Recall that a Salem measure is a measure whose Fourier transform decays as fast as its Hausdorff dimension allows, see Section 14.1. We prove that a class of measures, which includes the natural measure on fractal percolation, are Salem measures when their dimension is ≤ 2 (and this is sharp), adding to the relatively small number of known examples of Salem measures.…”

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confidence: 99%

Spatially independent martingales, intersections, and applications

Shmerkin

Суомала

2018

Memoirs of the AMS

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We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {η t } t , and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.

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confidence: 99%

Spatially independent martingales, intersections, and applications

Shmerkin

Суомала

2018

Memoirs of the AMS

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“…, where c = c(n) > 0 is the constant from Lemma 2.4 (1). Together with the definition of A i this implies…”

Section: Dimension Of General Measures On Large Conesmentioning

confidence: 74%

Local conical dimensions for measures

Ding¹,

Käenmäki²,

Суомала³

2013

Math. Proc. Camb. Phil. Soc.

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“…(1) Bandt and Kravchenko showed that there are plenty of C 1 planar self-affine curves (i.e. self-affine sets that are C 1 planar curves); see [1,Theorem 2]. Furthermore, in [1,Theorem 3(ii)], they showed that parabolic arcs and straight line segments are the only simple C 2 planar self-affine curves.…”

Section: Self-affine Sets and Analytic Curvesmentioning

confidence: 99%

“…self-affine sets that are C 1 planar curves); see [1,Theorem 2]. Furthermore, in [1,Theorem 3(ii)], they showed that parabolic arcs and straight line segments are the only simple C 2 planar self-affine curves. This result also follows from Theorem A by a simple modification.…”

Section: Self-affine Sets and Analytic Curvesmentioning

confidence: 99%