Search arXiv⌕ Search

arXiv subjects

Kenneth Falconer

Publications and source records attributed to Kenneth Falconer.

17 recordsLinked to original sources
arXiv2026

The sequence property for fractal dimensions

For various definitions of fractal dimension that are finitely stable, including upper box dimension, upper intermediate dimensions and Assouad spectra, we show that, given a compact subset $E$ of a metric space, typically $R^n$, there is a convergent sequence of points contained in $E$ of the same dimension as $E$ itself. Moreover, under certain conditions it is possible for a sequence to witness the dimension of $E$ for many definitions of dimension simultaneously, for example in a self-affine set $E$ there is a single convergent sequence that has the same Assouad spectrum or intermediate dimension values as $E$ itself.

math.MG↗
arXiv2024

Fractal percolation on statistically self-affine carpets

We consider a random self-affine carpet $F$ based on an $n\times m$ subdivision of rectangles and a probability $0<p<1$. Starting by dividing $[0,1]^2$ into an $n\times m$ grid of rectangles and selecting these independently with probability $p$, we then divide the selected rectangles into $n\times m$ subrectangles which are again selected with probability $p$; we continue in this way to obtain a statistically self-affine set $F$. We are particularly interested in topological properties of $F$. We show that the critical value of $p$ above which there is a positive probability that $F$ connects the left and right edges of $[0,1]^2$ is the same as the critical value for $F$ to connect the top and bottom edges of $[0,1]^2$. Once this is established we derive various topological properties of $F$ analogous to those known for self-similar carpets.

math.MG↗
arXiv2021

Intersections of thick compact sets in $\mathbb{R}^d$

We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in $\mathbb{R}^d$ with thickness $\tau$, there is a number $N(\tau)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(\tau)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

math.CA↗
arXiv2020

The density of sets containing large similar copies of finite sets

We prove that if $E \subseteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-measurable set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.

math.CA↗
arXiv2016

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Given a measure $\nu$ on a regular planar domain $D$, the Gaussian multiplicative chaos measure of $\nu$ studied in this paper is the random measure ${\widetilde \nu}$ obtained as the limit of the exponential of the $\gamma$-parameter circle averages of the Gaussian free field on $D$ weighted by $\nu$. We investigate the dimensional and geometric properties of these random measures. We first show that if $\nu$ is a finite Borel measure on $D$ with exact dimension $\alpha>0$, then the associated GMC measure ${\widetilde \nu}$ is non-degenerate and is almost surely exact dimensional with dimension $\alpha-\frac{\gamma^2}{2}$, provided $\frac{\gamma^2}{2}<\alpha$. We then show that if $\nu_t$ is a H\"{o}lder-continuously parameterized family of measures then the total mass of ${\widetilde \nu}_t$ varies H\"{o}lder-continuously with $t$, provided that $\gamma$ is sufficiently small. As an application we show that if $\gamma<0.28$, then, almost surely, the orthogonal projections of the $\gamma$-Liouville quantum gravity measure ${\widetilde \mu}$ on a rotund convex domain $D$ in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with H\"{o}lder continuous densities. Furthermore, ${\widetilde \mu}$ has positive Fourier dimension almost surely.

math.PR↗
arXiv2015

The dimension of projections of self-affine sets and measures

Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if mu is a Bernoulli measure on E with dim_H mu = dim_L mu, where dim_H and dim_L denote Hausdorff and Lyapunov dimensions, then the projection of mu in all but at most one direction has Hausdorff dimension min{dim_H mu,1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dim_H E,1} in all but at most one direction.

math.DS↗
arXiv2015

Planar self-affine sets with equal Hausdorff, box and affinity dimensions

Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.

math.DS↗
arXiv2015

Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.

math.MG↗
arXiv2014

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.

math.MG↗
arXiv2014

Codimension formulae for the intersection of fractal subsets of Cantor spaces

We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\mathcal{C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\max\{\dim E +\dim F -\dim \mathcal{C}^m, 0\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.

math.MG↗
arXiv2014

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 $\mathbb{R}^2$ with Hausdorff dimension $\dim_H K >1$ such that the rotational components of the underlying similarities generate the full rotation group. Then for all $\epsilon >0$, writing $\pi_\theta$ for projection onto the line $L_\theta$ in direction $\theta$, the Hausdorff dimensions of the sections satisfy $\dim_H (K\cap \pi_\theta^{-1}x)> \dim_H K - 1 - \epsilon$ for a set of $x \in L_\theta$ of positive Lebesgue measure, for all directions $\theta$ 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.

math.PR↗
arXiv2012

Generalized dimensions of images of measures under Gaussian processes

We show that for certain Gaussian random processes and fields X:R^N to R^d, D_q(mu_X) = min{d, D_q(mu)/alpha} a.s. for an index alpha which depends on Holder properties and strong local nondeterminism of X, where q>1, where D_q denotes generalized q-dimension and where mu_X is the image of the measure mu under X. In particular this holds for index-alpha fractional Brownian motion, for fractional Riesz-Bessel motions and for certain infinity scale fractional Brownian motions.

math.PR↗
arXiv2012

Exact dimensionality and projections of random self-similar measures and sets

We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu's result \cite{FeHu09} for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures \cite{HoSh12} to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, $C^1$-images and distance sets.

math.DS↗
arXiv2011

Growth rate of an endomorphism of a group

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient.We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.

math.GR↗
arXiv2010

Multistable processes and localisability

We use characteristic functions to construct alpha(x)-multistable measures and integrals, where the measures behave locally like alpha-stable measures, but with the stability index alpha(x) varying with time x. This enables us to construct alpha(x)-multistable processes on R, that is processes whose scaling limit at time x is an alpha(x)-stable process. We present several examples of such multistable processes and examine their localisability.

math.PR↗
arXiv2009

Localisable moving average stable and multistable processes

We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a ``tangent process'', in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein-Uhlenbeck process and the multistable reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is, at each time t, a Lévy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations.

math.PR↗