Bohr chaoticity and Khintchin conjecture

The Sarnak conjecture, which concerns with the Birkhoff averages weighted by the Möbius sequence, asserts that all zero entropy systems are orthogonal to the Möbius sequence. Which systems are orthogonal to none of non-trivial weights? We define such systems as Bohr chaotic systems. The Bohr chaoticity is a complexity of the system and is a topological invariant; it implies the positivity of entropy. However, the positivity of entropy doesn’t imply the Bohr chaoticity. We prove that a system $(X, T)$ admitting a horseshoe (i.e a susbsytem of some power of $T$ is conjugate to a full shift) is Bohr chaotic. Thus the usual nice systems of positive entropy are Bohr chaotic. But systems having few ergodic measures are not Bohr chaotic. Another class of systems which are proved to be Bohr chaotic are the algebraic principal systems. These are joint works with Shilei FAN (Wuhan), Valery RYZHYKOV (Moscou), Klaus SCHMIDT (Vienna), Weixiao SHEN (Shanghai) and Evgeny VERBITSKIY (Leiden). Also I would like to talk about Khintchin’s conjecture, a related problem in a setting of actions of mutiplicative semigroups of integers (more generally, actions of surjective endomorphisms of a compact Abelian group). But there is more questions than results for this topic.