Ubiquitous systems and metric number theory

Ubiquitous Systems and Metric Number Theory

We investigate the size and large intersection properties of Et = {x ∈ R d | x − k − x i < r i t for infinitely many (i, k) ∈ I µ,α × Z d }, where d ∈ N, t ≥ 1, I is a denumerable set, (x i , r i) i∈I is a family in [0, 1] d × (0, ∞) and I µ,α denotes the set of all i ∈ I such that the µ-mass of the ball with center x i and radius r i behaves as r i α for a given Borel measure µ and a given α >...

Metric number theory, lacunary series and systems of dilated functions

Contributions to Metric Number Theory

The aim of this work is to investigate some arithmetical properties of real numbers, for example by considering sequences of the type ([bα]) , n = 1, 2, . . . where b ∈ N, α ∈ R, the terms of the sequences being in arithmetical progression, square-free, sums of two squares or primes. The results are most commonly proved for almost all α ∈ R or (α1, . . . , αm) ∈ R m (in the sense of Lebesgue me...

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming from dynamics on homogeneous spaces. Glossary 1. Definition: Metric Diophantine approximation 2. Basic facts 3. Introduction 4. Connection with dynamics on the space of lattices 5. Diophantine app...

Control Systems and Number Theory

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, th...