Dimension and projections in normed spaces and Riemannian manifolds

This thesis is concerned with the behavior of Hausdorff measure and Hausdorff dimension under projections. In 1954, Marstrand proved that given a Borel set A Ϲ R² of dimension strictly larger than 1, for almost every line L that passes through the origin, the orthogonal projection of A onto L is a set of positive Hausdorff 1-measure. This theorem marked the start of a long sequence of results in the same spirit that are nowadays known as Marstrand-type projection theorems. In the first part of this thesis, we establish Marstrand-type projection theorems for projections induced by linear foliations as well as for closest-point projections onto hyperplanes in finite dimensional normed spaces. By the same methods we obtain a Besicovitch-Federer-type characterization of purely unrectifiable sets in terms of these families of projections. Moreover, we give an example underlining the sharpness of our results. In the second part of the thesis, we establish Marstrand-type as well as Besicovitch-Federer-type projection theorems for orthogonal projections along geodesics in hyperbolic space as well as in the two-sphere. Several of these results are achievable by two different methods: potential theoretic methods and Fourier analytic methods. We discuss the scope of each of these methods in both settings

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.Comment: 6 pages, 2 figure

An uncountable ergodic Roth theorem and applications

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This extends a previous result of Bergelson, McCutcheon and Zhang. Using this uncountable Roth theorem, we establish the following two additional results. [(i)] We establish a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. [(ii)] We establish a uniform bound on the lower Banach density of the set of double recurrence times along all $\Gamma$-systems, where $\Gamma$ is any group in a class of uniformly amenable groups. As a special case, we obtain this uniformity over all $\mathbb{Z}$-systems, and our result seems to be novel already in this particular case. Our uncountable Roth theorem is crucial in the proof of both of these results.Comment: 34 pages, [v2]: typos corrected, [v3]: improved presentation following referee's feedback, title and abstract change

Ongoing toxin-positive diphtheria outbreaks in a federal asylum centre in Switzerland, analysis July to September 2022.

Two diphtheria outbreaks occurred in a Swiss asylum center from July to October 2022, one is still ongoing. Outbreaks mainly involved minors and included six symptomatic respiratory diphtheria cases requiring antitoxin. Phylogenomic analyses showed evidence of imported and local transmissions of toxigenic strains in respiratory and skin lesion samples. Given the number of cases (n = 20) and the large genetic diversity accumulating in one centre, increased awareness and changes in public health measures are required to prevent and control diphtheria outbreaks

Dimension and projections in normed spaces and Riemannian manifolds

This thesis is concerned with the behavior of Hausdorff measure and Hausdorff dimension under projections. In 1954, Marstrand proved that given a Borel set A Ϲ R² of dimension strictly larger than 1, for almost every line L that passes through the origin, the orthogonal projection of A onto L is a set of positive Hausdorff 1-measure. This theorem marked the start of a long sequence of results in the same spirit that are nowadays known as Marstrand-type projection theorems. In the first part of this thesis, we establish Marstrand-type projection theorems for projections induced by linear foliations as well as for closest-point projections onto hyperplanes in finite dimensional normed spaces. By the same methods we obtain a Besicovitch-Federer-type characterization of purely unrectifiable sets in terms of these families of projections. Moreover, we give an example underlining the sharpness of our results. In the second part of the thesis, we establish Marstrand-type as well as Besicovitch-Federer-type projection theorems for orthogonal projections along geodesics in hyperbolic space as well as in the two-sphere. Several of these results are achievable by two different methods: potential theoretic methods and Fourier analytic methods. We discuss the scope of each of these methods in both settings

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections

Projection theorems for linear-fractional families of projections

Marstrand's theorem states that applying a generic rotation to a planar set $A$ before projecting it orthogonally to the $x$-axis almost surely gives an image with the maximal possible dimension $\min(1, \dim A)$. We first prove, using the transversality theory of Peres-Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in PSL(2,\C) or a generic real linear-fractional transformation in $PGL(3,\R)$. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of PSL(2,\C) or $PGL(3,\R)$. Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries

Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand's theorem, Kaufman's theorem, and Falconer's theorem in the above geometrical settings

Marstrand type projection theorems for normed spaces

We consider Marstrand type projection theorems for closest-point projections in the normed space ℝ². We prove that if a norm on ℝ² is regular enough, then the analogues of the well-known statements from the Euclidean setting hold, while they fail for norms whose unit balls have corners. We establish our results by verifying Peres and Schlag’s transversality property and thereby also obtain a Besicovitch-Federer type characterization of purely unrectifiable set