Mass Transference Principles and Applications in Diophantine Approximation

This thesis is concerned with the Mass Transference Principle and its applications in Diophantine approximation. The Mass Transference Principle, proved by Beresnevich and Velani in 2006, is a powerful result allowing for the transference of Lebesgue measure statements for $\limsup$ sets arising from sequences of balls in $\mathbb{R}^k$ to Hausdorff measure statements. The significance of this result is especially prominent in Diophantine approximation, where many sets of interest arise naturally as $\limsup$ sets. We establish a general form of the Mass Transference Principle for systems of linear forms conjectured by Beresnevich, Bernik, Dodson and Velani in 2009. This improves upon an earlier result in this direction due to Beresnevich and Velani from 2006. In addition, we present a number of applications of this ``new'' mass transference principle for linear forms to problems in Diophantine approximation, some of which were previously out of reach when using the result of Beresnevich and Velani. These include a general transference of Lebesgue measure Khintchine--Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine--Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira. Using a Hausdorff measure analogue of the inhomogeneous Khintchine--Groshev Theorem (established via the mass transference principle for linear forms), we give an alternative proof of most cases of a general inhomogeneous Jarn\'{\i}k--Besicovitch Theorem which was originally proved by Levesley in 1998. We additionally show that without monotonicity Levesley's theorem no longer holds in general. We conclude this thesis by discussing the concept of a mass transference principle for rectangles. In particular, we demonstrate how some known results may be extended using a slicing technique

Capacity of elements of Banach algebras

As its name suggests, this thesis is an account of the recent theory of the capacity of elements of Banach algebras. The first chapter contains a summary of the background theory, other than fundamentals, used, and consists mainly of perturbation theory of linear operators and certain properties (Jf strictly singular operators. This chapter relies heavily on the work of T„ Kato, both in his own papers and the book by S. Goldberg "Unbounded Linear Operators". Chapter 2 introduces the notion of capacity, following Halmos in his paper "Capacity in Banach algebras", and several small new results are proved, and counterexamples given, to tidy up "loose ends". The question of the capacity of the sum of two quasialgebraic elements (i.e. ones with capacity zero) is raised, and a partial solution given. The perturbation theory of Chapter 1 is applied to show the equality of the capacity of the spectrum and the Fredholn spectrum of an operator on a Banach space, whence it is shown that if J is a closed two-sided ideal of B(x) containing only Riesz operators, then perturbation by an element of J leaves the capacity invariant; this is true, in particular, for compact operators. A converse theorem is proved for Hilbert space, Chapter 3 introduces the new concept of the joint capacity of an r-tuple of elements cf a commutative Banach algebra, and develops the theory of this notion, Much of the theory parallels, xn a weaker form, that of the original concept, but there are significant differences. Finally, a perturbation theorem, similar to the original one is proved for the joint capacity

Distribution of additive functions in algebraic number fields

1. Background. This thesis is based on ideas drawn from classical probabilistic number theory, from the work of Novoselov [2], and from the relevant work on algebraic number fields. Classical probabilistic number theory (as described in Elliott [1], for example) is concerned with the distribution of arithmetic functions on the ring of (rational) integers, ZZ. Two well-known results in this area are the Hardy-Ramanujan and the Erdos-Wintner theorems. The Hardy-Ramanujan theorem states that, in some sense, every integer n has about log log n prime divisors, and the Erdos-Wintner theorem gives conditions under·which additive functions have limiting distributions. The original proofs of both results were subsequently considerably simplified by using a result known as the Tunin-Kubilius inequality. Although results in this field have a definite probabilistic flavour, it has not proved easy to establish them by a direct appeal to the theory of probability. Novoselov [2] developed a probability space which provides a natural framework for developing results of probabilistic number theory from results of probability. For example, using standard results from probability theory and some arithmetic estimates (which amount to the Turan-Kubilius inequality) he obtained the Hardy-Ramanujan and Erdos-Wintner theorems. Many of the results of probabilistic number theory have been generalized to results concerning the distribution of additive functions on the ideals of the ring, V, of integers of an algebraic number field (see Prachar [3], for example). However, work in this area has not used a probabilistic framework as fully as in the classical case of ZZ. 2. Aims. The aim of this thesis is to set up a space for probabilistic number theory in algebraic number fields analogous to that of Novoselov [2] for ZZ and to apply his approach to develop analogues in 1) of the Hardy-Ramanujan and Erdos-Wintner theorems. We endeavour to produce as much as possible without the use of sieve results. 3. Contents Chapter 1 of this thesis is an introduction to the background outlined above, and Chapter 2 gathers together some preliminary material. In Chapter 3 we obtain an analogue of the Tunin-Kubilius inequality in D. For this purpose we estimate the number of elements of an ideal which lie in a multiple of the fundamental domain of D (viewed as a lattice). In Chapter 4 we construct a probability space Ω, containing D, using two different approaches. One approach is analogous to that in Novoselov [2). The other views n as the product of the completions of D with respect to its non-Archimedean valuations and this enables us to simplify some proofs. rn· Chapter 5 we prove versions of the Hardy-Ramanujan and ErdosWintner theorems for additive functions on the principal ideals of D. Some examples are discussed. In Chapter 6 we consider additive functions on all the ideals of 1) (not just the principal ideals). We prove Prachar's version of the Hardy-Ramanujan theorem (see Prachar [3]) by using the results of Chapter 5 and the correspondence between the ideals of D in a given class and certain elements of V. References [1] Elliott,P.D.T.A., Probabilistic number theory, Volumes I and 11. Springer-Verlag, New York (1979). [2] Novoselov,E.V., A new method in probabilistic number theory. Amer. Math. Soc. Translations (2) 52 (1966), pp. 217-275. (3] Prachar,K., Verallgemeinerung eines Satzes 'IJOn Hardy und Ramanujan auf algebraische Zahlkorper. Monatsh. Math. 56 (1952), pp. 229-232.Thesis (MSc) -- University of Adelaide, Dept. of Pure Mathematics, 198

Algebraic groups over the adeles

Die vorliegende Arbeit beschäftigt sich mit algebraischen Gruppen G definiert über einem algebraischen Zahlkörper k. Um Informationen über die k-Punkte von G zu erhalten können wir die adelischen Punkte G_A von G betrachten. Ein Ziel dieser Arbeit ist die Konstruktion von Fundamentalbereichen bzw. -mengen für G_k in G_A. Weiters sollen Kriterien für Kompaktheit sowie für endliches invarinates Volumen des Quotienten G_A/G_k gefunden werden. Ferner betrachten wir die Gruppe G_A^(1), definiert als der Schnitt über die Kerne aller k-Charaktere von G, und wollen auch hier Bedingungen finden, die garantieren, dass der Quotient G_A^(1)/G_k kompakt ist bzw. endliches invariantes Volumen hat. Am Ende dieser Arbeit betrachten wir Inklusionen i: H -> G von reduktiven algebraischen k-Gruppen und zeigen, dass der induzierte Morphismus i: H_A^(1)/H_k -> G_A^(1)/G_k eigentlich ist.This diploma thesis deals with algebraic groups G defined over algebraic number fields k. To gain information about the k-points of G we can consider the adelic points G_A of G. One aim of this thesis is to construct fundamental domains respectively sets for G_k in G_A. In addition, criteria for compactness and for the existence of a finite invariant volume for the quotient G_A/G_k shall be found. Furthermore, we consider the group G_A^(1) defined by the intersection of the kernels of all k-characters of G and again try to find conditions which guarantee that the quotient G_A^(1)/G_k is compact, has finite invariant volume respectively. At the end of this thesis we analyse inclusions i: H -> G of reductive algebraic k-groups and show that the induced morphism i: H_A^(1)/H_k -> G_A^(1)/G_k is proper

Application of Probability Methods in Number Theory and Integral Geometry

Gusakova A. Application of Probability Methods in Number Theory and Integral Geometry. Bielefeld: Universität Bielefeld; 2018

Cognitive-developmental learning for a humanoid robot : a caregiver's gift

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 319-341).(cont.) which are then applied to developmentally acquire new object representations. The humanoid robot therefore sees the world through the caregiver's eyes. Building an artificial humanoid robot's brain, even at an infant's cognitive level, has been a long quest which still lies only in the realm of our imagination. Our efforts towards such a dimly imaginable task are developed according to two alternate and complementary views: cognitive and developmental.The goal of this work is to build a cognitive system for the humanoid robot, Cog, that exploits human caregivers as catalysts to perceive and learn about actions, objects, scenes, people, and the robot itself. This thesis addresses a broad spectrum of machine learning problems across several categorization levels. Actions by embodied agents are used to automatically generate training data for the learning mechanisms, so that the robot develops categorization autonomously. Taking inspiration from the human brain, a framework of algorithms and methodologies was implemented to emulate different cognitive capabilities on the humanoid robot Cog. This framework is effectively applied to a collection of AI, computer vision, and signal processing problems. Cognitive capabilities of the humanoid robot are developmentally created, starting from infant-like abilities for detecting, segmenting, and recognizing percepts over multiple sensing modalities. Human caregivers provide a helping hand for communicating such information to the robot. This is done by actions that create meaningful events (by changing the world in which the robot is situated) thus inducing the "compliant perception" of objects from these human-robot interactions. Self-exploration of the world extends the robot's knowledge concerning object properties. This thesis argues for enculturating humanoid robots using infant development as a metaphor for building a humanoid robot's cognitive abilities. A human caregiver redesigns a humanoid's brain by teaching the humanoid robot as she would teach a child, using children's learning aids such as books, drawing boards, or other cognitive artifacts. Multi-modal object properties are learned using these tools and inserted into several recognition schemes,by Artur Miguel Do Amaral Arsenio.Ph.D