# Sascha TroscheitUppsala University | UU · Department of Mathematics

Sascha Troscheit

Doctor of Philosophy

## About

42

Publications

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355

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Introduction

Additional affiliations

June 2020 - present

January 2019 - June 2020

May 2017 - December 2018

## Publications

Publications (42)

We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such sets has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-c...

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the 'upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain,...

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.

The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similar...

We investigate the box-counting dimension of the image of a set E⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}$$\end{document} under a random...

In his monograph on Infinite Abelian Groups, I. Kaplansky raised three “test problems” concerning their structure and multiplicity. As noted by Azoff, these problems make sense for any category admitting a direct sum operation. Here, we are interested in the operator theoretic version of Kaplansky’s second problem which asks: if A and B are operato...

A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem...

The $\phi$-Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key prop...

In his monograph on Infinite Abelian Groups, I. Kaplansky raised three ``test problems" concerning their structure and multiplicity. As noted by Azoff, these problems make sense for any category admitting a direct sum operation. Here, we are interested in the operator theoretic version of Kaplansky's second problem which asks: if $A$ and $B$ are op...

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have a well-defined Minkowski...

In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to bound the Hausdorff dimension of cylindrical shrinking target and recurrence sets. For generic self-affine sets i...

We investigate the box-counting dimension of the image of a set $E \subset \mathbb{R}$ under a random multiplicative cascade function $f$. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for box-counting dimension....

In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to bound the Hausdorff dimension of cylindrical shrinking target and recurrence sets. For generic self-affine sets i...

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log comensurable contraction ratios, have a well-defined Minkowski...

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a dete...

The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which we call the Assouad dimension (of the measure) and is also known as the upper regularity dimension. One reason for the interest in this notion is t...

The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, less abstract, metric spaces. Similar...

The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by considering the $L^p$ norms of its density. We establish sharp relationships between these two notions. Roughly...

A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating set $X$ is a generating set such that no proper subset of $X$ is also a generating set. A semigroup $S$ is ubiquitous if every irredundant generating set of $S$ is of the same cardinality. We are motivated by a na\"{i}ve al...

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular,...

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ i...

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in ra...

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular,...

The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and o...

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of random self-affine Bedford-McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely `as large as possible' (a deterministic qua...

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain,...

We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the deterministic case. The overlapping case is more complicated; we introduce the notion of finite type for random homoge...

Random code-trees with necks were introduced recently to generalise the notion of $V$-variable and random homogeneous sets. While it is known that the Hausdorff and packing dimensions coincide irrespective of overlaps, their exact Hausdorff and packing measure has so far been largely ignored. In this article we consider the general question of an a...

Vlasov-Maxwell equilibria are described by the self-consistent solutions of the time-independent Maxwell equations for the real-space dynamics of electromagnetic fields, and the Vlasov equation for the phase-space dynamics of particle distributions in a collisionless plasma. These two systems (macroscopic and microscopic) are coupled via the source...

The class of stochastically self-similar sets contains many famous examples of random sets, e.g. Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive set...

In this article we discuss the Mass Transference Principle due to Beresnevich and Velani and survey several generalisations and variants, both deterministic and random. Using a Hausdorff measure analogue of the inhomogeneous Khintchine-Groshev Theorem, proved recently via an extension of the Mass Transference Principle to systems of linear forms, w...

We present the solution to an inverse problem arising in the context of
finding a distribution function for a specific collisionless plasma
equilibrium. The inverse problem involves the solution of two integral
equations, each having the form of a Weierstrass transform. We prove that
inverting the Weierstrass transform using Hermite polynomials lea...

We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov-Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Max...

We show that self-conformal subsets of $\mathbb{R}$ that do not satisfy the weak separation condition have full Assouad dimension. Combining this with a recent results by K\"aenm\"aki and Rossi we conclude that an interesting dichotomy applies to self-conformal and not just self-similar sets: if $F\subset\mathbb{R}$ is self-conformal with Hausdorff...

In this paper we study two random analogues of the box-like self-affine
attractors introduced by Fraser, itself an extension of Sierpi\'nski carpets.
We determine the almost sure box-counting dimension for the homogeneous random
case ($1$-variable random), and give a sufficient condition for the almost sure
box dimension to be the expectation of th...

In this paper we propose a new model of random graph directed fractals that
extends the current well-known model of random graph directed iterated function
systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We
study its dimensional properties for similarities with and without overlaps. In
particular we show that for the two c...

We present a first discussion and analysis of the physical properties of a
new exact collisionless equilibrium for a one-dimensional nonlinear force-free
magnetic field, namely the Force-Free Harris Sheet. The solution allows any
value of the plasma beta, and crucially below unity, which previous nonlinear
force-free collisionless equilibria could...

We consider several different models for generating random fractals including
random self-similar sets, random self-affine carpets, and fractal percolation.
In each setting we compute either the almost sure or the Baire typical Assouad
dimension and consider some illustrative examples. Our results reveal a common
phenomenon in all of our models: th...

In this paper we consider the probability distribution function of a Gibbs
measure supported on a self-conformal set given by an iterated function system
(devil's staircase). We use thermodynamic multifractal formalism to calculate
the Hausdorff dimension of the sets $S^{\alpha}_{0}$, $S^{\alpha}_{\infty}$ and
$S^{\alpha}$, the set of points at whi...