We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdös-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak's Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,x). The talk is based on recent joint work with Florian Richter.[-]

Discussing recent work joint with M. Rudnev [2], I will discuss the modern approach to the sum-product problem in the reals. Our approach builds upon and simplifies the arguments of Shkredov and Konyagin [1], and in doing so yields a new best result towards the problem. We prove that
$max(\left | A+A \right |,\left | A+A \right |)\geq \left | A \right |^{\frac{4}{3}+\frac{2}{1167}-o^{(1)}}$ , for a finite $A\subset \mathbb{R}$. At the heart of our argument are quantitative forms of the two slogans ‘multiplicative structure of a set gives additive information', and ‘every set has a multiplicatively structured subset'.[-]