# Intermediate dimensions

@article{Falconer2018IntermediateD, title={Intermediate dimensions}, author={Kenneth J. Falconer and Jonathan M. Fraser and Tom Kempton}, journal={Mathematische Zeitschrift}, year={2018}, volume={296}, pages={813-830} }

We introduce a continuum of dimensions which are ‘intermediate’ between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $$|U| \le |V|^\theta $$ | U | ≤ | V | θ for all sets U , V used in a particular cover, where $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] is a parameter. Thus, when $$\theta =1$$ θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions…

## 21 Citations

### Generalised intermediate dimensions

- Mathematics
- 2020

We introduce a family of dimensions, which we call the $\Phi$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by…

### On the intermediate dimensions of Bedford-McMullen carpets

- Mathematics
- 2020

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a…

### Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

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- 2021

Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the…

### Dimensions of popcorn-like pyramid sets

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This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the…

### Interpolating Between Dimensions

- Computer ScienceFractal Geometry and Stochastics VI
- 2021

A new approach in dimension theory is surveyed, which seeks to unify the study of individual dimensions by viewing them as different facets of the same object.

### Attainable forms of intermediate dimensions

- MathematicsAnnales Fennici Mathematici
- 2022

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function…

### Dimensions of Fractional Brownian Images

- MathematicsJournal of Theoretical Probability
- 2021

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds…

### Intermediate Dimensions: A Survey

- MathematicsLecture Notes in Mathematics
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This article surveys the $\theta$-intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when $\theta=0$ to…

### Box-counting measure of metric spaces

- Mathematics
- 2021

. In this paper, we introduce a new notion called the box-counting measure of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure;…

### Assouad type dimensions of infinitely generated self-conformal sets

- Mathematics
- 2022

. We study the dimension theory of limit sets of iterated function systems consisting of a countably inﬁnite number of conformal contractions. Our focus is on the Assouad type dimensions, which give…

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