Diophantine Approximation and Dirichlet Series

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. A Review of Commutative Harmonic Analysis

   Hervé Queffelec, Martine Queffelec

   Abstract

   This chapter might be skipped at first reading. But we have the feeling that a minimal knowledge of basic facts in harmonic analysis is necessary to understand certain aspects of the analytic theory of Dirichlet series, especially those connected with almost-periodicity, ergodic theory, the Bohr point of view to be developed later, and also universality problems.

3. Chapter 2. Ergodic Theory and Kronecker’s Theorems

   Hervé Queffélec, Martine Queffélec

   Abstract

   Measure theory, sometimes, brings out the existence of specific elements by giving a positive measure to the set of such objects. For want of anything better, it can also be used in the number-theoretical framework to produce classifications of real numbers through their expansions. Ergodic theory will play a role in this purpose.

4. Chapter 3. Diophantine Approximation

   Hervé Queffélec, Martine Queffélec

   Abstract

   Throughout this chapter, [x] denotes the integral part and \(\{x\}\) the fractional part of the real number x so that \(x = [x] + \{x\}\); moreover we shall use the notation \(\Vert x\Vert \) for the closest distance of x to an element of \(\mathbb {Z}\).

5. Chapter 4. General Properties of Dirichlet Series

   Hervé Queffélec, Martine Queffélec

   Abstract

   For a real number \(\theta \), denote  \(\mathbb {C}_{\theta }\) the set of complex numbers whose real part exceeds \(\theta \).

6. Chapter 5. Probabilistic Methods for Dirichlet Series

   Hervé Queffélec, Martine Queffélec

   Abstract

   The title of this chapter is a little emphatic, because the probabilistic methods will here concentrate essentially on one maximal inequality, which is fairly well-known in harmonic analysis, but will have a specific aspect, due to the Bohr point of view on Dirichlet series.

7. Chapter 6. Hardy Spaces of Dirichlet Series

   Hervé Queffélec, Martine Queffélec

   Abstract

   The forthcoming spaces \(\mathcal {H}^{p}\) of Dirichlet series \((1 \le p \le \infty )\), analogous to the familiar Hardy spaces \(H^{p}\) on the unit disk, have been successfully introduced to study completeness problems in Hilbert spaces [1], first for \(p = 2, \infty \). Later on, the general case was considered in [2] for the study of composition operators.

8. Chapter 7. Voronin-Type Theorems

   Hervé Queffélec, Martine Queffélec

   Abstract

   In this introductory section, we begin by fixing some notations, recalling some basic facts on Dirichlet characters [1, Chap. 5] and presenting the main results to be discussed. The techniques (Hilbertian spaces of analytic functions, ergodic theorems) are a good illustration of the material introduced in the previous chapters.

9. Chapter 8. Composition Operators on the Space of Dirichlet Series

   Hervé Queffélec, Martine Queffélec

   Abstract

   The general framework for composition operators acting on a Banach space X of functions analytic on a domain \(\varOmega \) of \( \mathbb {C}^d,\ 1\le d\le \infty \) is the following: we always assume that X is continuously embedded in the Fréchet space \(H(\varOmega ):=\hbox { Hol}\ (\varOmega )\), so that the point evaluations \(\delta _a, \ \delta _{a}(f)=f(a)\) are continuous linear forms on X for each \(a\in \varOmega \).

10. Backmatter