Projective metric number theory

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...

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 >...

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...

Metric number theory, lacunary series and systems of dilated functions

Metric Connections in Projective Differential Geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikeš, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sec...