The Logarithmic Sarnak Conjecture for Countably Ergodic Systems

Abstract

The main goal of this thesis is to provide a roughly self-contained account of Frantzikinakis and Host’s proof of the logarithmic Sarnak conjecture for topological dynamical systems with zero topological entropy and countably many ergodic invariant measures. This beautiful result combines classical ideas introduced by Furstenberg in his famous proof of Szemerédi’s theorem, recent developments in analytic number theory, including results on two-point correlations of the Möbius function, due to Tao, and a key identity that arises from his novel entropy decrement argument, and structural classification theorems in ergodic theory, building upon work by the two main authors and many others. Each of the first three chapters is dedicated to one of these topics, and the proofs of the main results are given in the last chapter, after all the necessary tools have been introduced.