We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

In this survey article we start from the famous Furstenberg theorem on non-lacunary semigroups of integers, and next we present its generalizations and some related results.

2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

The Littlewood conjecture in Diophantine approximation claims that$$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·\parallel q\beta \parallel =0$$holds for all real numbers $\alpha$ and $\beta$, where $\parallel ·\parallel$ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that$$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·{\left|q\right|}_p=0$$holds for every real number $\alpha$ and every prime number $p$, where ${|·|}_p$ denotes the $p$-adic absolute value normalized by ${\left|p\right|}_p=p^{-1}$. We survey the known results on these conjectures and highlight recent developments.