Shrinking targets on Bedford-McMullen carpets

Bárány, Balázs; Rams, Michał

doi:10.1112/plms.12151

Cited by 14 publications

(6 citation statements)

References 28 publications

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“…In recent years, several other authors have also studied the problem of shrinking targets on fractals. For example, Chernov and Kleinbock [9] studied the measure of shrinking target sets with respect to ergodic measures, Chang, Wu, and Wu [8] very recently studied the problem of recurrence sets on linear IFSs consisting of maps with equal contraction ratios, Koivusalo and Ramírez [25] considered shrinking targets on self-affine sets, the second author and Rams computed the Hausdorff dimension for certain shrinking targets on Bedford-McMullen carpets [4], and Seuret and Wang considered some related problems in the setting of conformal IFSs [32]. However, the value of the dim H (W (x, ))-dimensional Hausdorff measure remained unknown except in some very special cases.…”

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confidence: 99%

On the Hausdorff Measure of Shrinking Target Sets on Self‐conformal Sets

Allen

Bárány

2021

Mathematika

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In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995 [Invent. Math. 119(1) (1995), 175-198]. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets".

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confidence: 99%

On the Hausdorff Measure of Shrinking Target Sets on Self‐conformal Sets

Allen

Bárány

2021

Mathematika

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“…In recent years, several other authors have also studied the problem of shrinking targets on fractals. For example, Chernov and Kleinbock [12] studied the measure of shrinking target sets with respect to ergodic measures, Chang, Wu and Wu [11] very recently studied the problem of shrinking targets on linear iterated iterated function systems consisting of maps with equal contraction ratios, Koivusalo and Ramírez [30] considered shrinking targets on self-affine sets, the second author and Rams computed the Hausdorff dimension for certain shrinking targets on Bedford-McMullen carpets [4], and Seuret and Wang considered some related problems in the setting of conformal iterated function systems [39].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the Hausdorff measure of shrinking target sets on self-conformal sets

Allen¹,

Bárány²

2019

Preprint

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Motivated by a classical question of Mahler (1984), Levesley, Salp, and Velani (2007) showed that the Hausdorff measure of the set of points in the middle-third Cantor set which are ψ-well-approximable by triadic rationals satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of this set is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on ψ.In this article, we prove an analogue of this result, obtaining a zero-full dichotomy for Hausdorff measure, in the setting of more general self-conformal sets with an appropriate adapted notion of approximation. Unlike in the work of Levesley, Salp, and Velani, we show that we are unable to apply the Mass Transference Principle due to Beresnevich and Velani (2006) in our setting. Instead, our proof relies on recasting the problem in the language of symbolic dynamics and appealing to several concepts from thermodynamic formalism, eventually enabling us to use an analogue of the mass distribution principle. In addition to demonstrating how our main result naturally extends the work of Levesley, Salp, and Velani, and complements some recent work of Baker (2018), we apply our main result to obtain a Jarník type statement for the Hausdorff measure of the set of badly approximable numbers which are "well-approximable" in some sense by fixed quadratic irrationals. This relies on the fact that the set of badly approximable numbers with partial quotients bounded above by a fixed M P N forms a self-conformal, but not self-similar, set. Hence this is a novel application of our main result which does not follow directly from previous results in this direction.

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“…The shrinking target problem has intricate links to number theory when using naturally arising sets in Diophantine approximation as the shrinking targets; e.g. see [AB21,BR18,PR17].…”

Section: Applicationsmentioning

confidence: 99%

Uniform quasi-multiplicativity of locally constant cocycles and applications

Mohammadpour¹,

Park²

2022

Preprint

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In this paper, we show that a locally constant cocycle A is k-quasi multiplicative under the irreducibility assumption. More precisely, we show that if A t and A ∧m are irreducible for every t | d and 1 ≤ m ≤ d − 1, then A is k-uniformly spannable for some k ∈ N, which implies that A is k-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is ψ-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets.

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Shrinking targets on Bedford-McMullen carpets

Abstract: We describe the shrinking target set for the Bedford–McMullen carpets, with targets being either cylinders or geometric balls.

Cited by 14 publications

References 28 publications

On the Hausdorff Measure of Shrinking Target Sets on Self‐conformal Sets

On the Hausdorff Measure of Shrinking Target Sets on Self‐conformal Sets

On the Hausdorff measure of shrinking target sets on self-conformal sets

Uniform quasi-multiplicativity of locally constant cocycles and applications

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