Intermediate dimension of images of sequences under fractional Brownian motion

We show that the almost sure θ-intermediate dimension of the image of the set Fp ={0, 1,1/2p,1/3p,...} under index-h fractional Brownian motion is θ/(ph+θ), a value that is smaller than that given by directly applying the Hölder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.PostprintPeer reviewe

Obituary - Richard Kenneth Guy, 1916-2020

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A capacity approach to box and packing dimensions of projections and other images

Dimension profiles were introduced by Falconer and Howroyd to provide formulae for the box-counting and packing dimensions of the orthogonal projections of a set E or a measure on Euclidean space onto almost all m-dimensional subspaces. The original definitions of dimension profiles are somewhat awkward and not easy to work with. Here we rework this theory with an alternative definition of dimension profiles in terms of capacities of E with respect to certain kernels, and this leads to the box-counting dimensions of projections and other images of sets relatively easily. We also discuss other uses of the profiles, such as the information they give on exceptional sets of projections and dimensions of images under certain stochastic processes. We end by relating this approach to packing dimension.Postprin

Intermediate dimensions - a survey

Funding: The work was supported in part by an EPSRC Standard Grant EP/R015104/1.This article surveys the θ-intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when θ=0 to box-counting dimensions when θ=1. We bring together diverse properties of intermediate dimensions which we illustrate by examples.Postprin

Dimension conservation for self-similar sets and fractal percolation

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let K be a self-similar subset of R2 with Hausdorff dimension dimHK >1 such that the rotational components of the underlying similarities generate the full rotation group. Then, for all ε >0, writing πθ for projection onto the Lθ in direction θ, the Hausdorff dimensions of the sections satisfy dimH (K ∩ πθ-1x)> dimHK - 1 - ε for a set of x ∈ Lθ of positive Lebesgue measure, for all directions θ except for those in a set of Hausdorff dimension 0. For a class of self-similar sets we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.PostprintPeer reviewe

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

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

A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set E in ℝn onto almost all m-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of E with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.PostprintPublisher PDFPeer reviewe

Assouad dimension influences the box and packing dimensions of orthogonal projections

Funding: UK EPSRC Standard Grant (EP/R015104/1) (KJF and JMF). Leverhulme Trust Research Project Grant (RPG-2019-034) (JMF). Royal Society International Exchange grant IES\R1\191195 (KJF and PS). ProjectPICT 2015-3675 (ANPCyT) (PS).We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ R n is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.Publisher PDFPeer reviewe

Minkowski dimension for measures

Funding: JMF and KJF are financially supported by an EPSRC Standard Grant (EP/R015104/1) and JMF by a Leverhulme Trust Research Project Grant (RPG-2019-034).The purpose of this article is to introduce and motivate the notion of Minkowski (or box) dimension for measures. The definition is simple and fills a gap in the existing literature on the dimension theory of measures. As the terminology suggests, we show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.PostprintPeer reviewe

Sixty years of fractal projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand’s projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Postprin