Box-counting dimension in one-dimensional random geometry of multiplicative cascades

Kenneth John Falconer; Sascha Troscheit

doi:10.1007/s00220-022-04558-9

Box-counting dimension in one-dimensional random geometry of multiplicative cascades

Kenneth John Falconer, Sascha Troscheit^*

^*Corresponding author for this work

Pure Mathematics

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Abstract

A result of Benjamini and Schramm shows that the Hausdorff dimension of sets in one-dimensional random geometry given by multiplicative cascades satisfies an elegant formula dependent only on the random variable and the dimension of the set in Euclidean geometry. In this article we show that this holds for the box-counting dimension when the set is sufficiently regular. This formula, however, is not valid in general and we provide general bounds on the box-counting dimension in the random metric. We explicitly compute the box-counting dimension for a large family of countable sets that accumulate at a single point which shows that the Benjamini-Schramm type formula cannot hold in general. This shows that the situation for the box-counting dimension is more subtle and knowledge of the structure is needed. We illustrate our results by providing examples including a pair of sets with the same box-counting dimension but different dimensions in the random metric.

Original language	English
Pages (from-to)	57–83
Number of pages	27
Journal	Communications in Mathematical Physics
Volume	399
Issue number	1
Early online date	18 Nov 2022
DOIs	https://doi.org/10.1007/s00220-022-04558-9
Publication status	Published - 1 Apr 2023

Keywords

Multiplicative cascades
Box-counting dimension
Random geometry

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Cite this

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title = "Box-counting dimension in one-dimensional random geometry of multiplicative cascades",

abstract = "A result of Benjamini and Schramm shows that the Hausdorff dimension of sets in one-dimensional random geometry given by multiplicative cascades satisfies an elegant formula dependent only on the random variable and the dimension of the set in Euclidean geometry. In this article we show that this holds for the box-counting dimension when the set is sufficiently regular. This formula, however, is not valid in general and we provide general bounds on the box-counting dimension in the random metric. We explicitly compute the box-counting dimension for a large family of countable sets that accumulate at a single point which shows that the Benjamini-Schramm type formula cannot hold in general. This shows that the situation for the box-counting dimension is more subtle and knowledge of the structure is needed. We illustrate our results by providing examples including a pair of sets with the same box-counting dimension but different dimensions in the random metric.",

keywords = "Multiplicative cascades, Box-counting dimension, Random geometry",

author = "Falconer, {Kenneth John} and Sascha Troscheit",

note = "Funding: ST was funded by Austrian Research Fund (FWF) Grant M-2813.",

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T1 - Box-counting dimension in one-dimensional random geometry of multiplicative cascades

AU - Falconer, Kenneth John

AU - Troscheit, Sascha

N1 - Funding: ST was funded by Austrian Research Fund (FWF) Grant M-2813.

PY - 2023/4/1

Y1 - 2023/4/1

N2 - A result of Benjamini and Schramm shows that the Hausdorff dimension of sets in one-dimensional random geometry given by multiplicative cascades satisfies an elegant formula dependent only on the random variable and the dimension of the set in Euclidean geometry. In this article we show that this holds for the box-counting dimension when the set is sufficiently regular. This formula, however, is not valid in general and we provide general bounds on the box-counting dimension in the random metric. We explicitly compute the box-counting dimension for a large family of countable sets that accumulate at a single point which shows that the Benjamini-Schramm type formula cannot hold in general. This shows that the situation for the box-counting dimension is more subtle and knowledge of the structure is needed. We illustrate our results by providing examples including a pair of sets with the same box-counting dimension but different dimensions in the random metric.

AB - A result of Benjamini and Schramm shows that the Hausdorff dimension of sets in one-dimensional random geometry given by multiplicative cascades satisfies an elegant formula dependent only on the random variable and the dimension of the set in Euclidean geometry. In this article we show that this holds for the box-counting dimension when the set is sufficiently regular. This formula, however, is not valid in general and we provide general bounds on the box-counting dimension in the random metric. We explicitly compute the box-counting dimension for a large family of countable sets that accumulate at a single point which shows that the Benjamini-Schramm type formula cannot hold in general. This shows that the situation for the box-counting dimension is more subtle and knowledge of the structure is needed. We illustrate our results by providing examples including a pair of sets with the same box-counting dimension but different dimensions in the random metric.

KW - Multiplicative cascades

KW - Box-counting dimension

KW - Random geometry

UR - https://arxiv.org/abs/2203.15315

U2 - 10.1007/s00220-022-04558-9

DO - 10.1007/s00220-022-04558-9

M3 - Article

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SP - 57

EP - 83

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -

Box-counting dimension in one-dimensional random geometry of multiplicative cascades

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Keywords

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