关于随机覆盖集的Ekström-Persson猜想

IF 1 2区数学 Q1 MATHEMATICS Journal of the London Mathematical Society-Second Series Pub Date : 2024-12-20 DOI:10.1112/jlms.70058

Esa Järvenpää, Maarit Järvenpää, Markus Myllyoja, Örjan Stenflo

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引用次数: 0

摘要

我们考虑了随机覆盖集的Hausdorff维数，这些随机覆盖集是由球组成的，这些球的中心是根据R d $\mathbb {R}^d$上的任意Borel概率度量随机选择的，半径由趋于零的确定性序列给定。我们证明，对于一定的参数范围，Ekström和Persson关于在半径（n−α） n =的特殊情况下维数精确值的猜想1∞$(n^{-\alpha })_{n=1}^\infty$。对于具有任意半径序列的球，我们找到了尺寸的明确界限，并证明了Ekström-Persson猜想的自然扩展在这种情况下是不成立的。最后，我们构造了实例，证明不存在只涉及测度的上下局部维数和由半径序列决定的关键参数的维数公式。

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The Ekström–Persson conjecture regarding random covering sets

We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on $R^{d}$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii ${(n^{- α})}_{n = 1}^{\infty}$ . For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström–Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.

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