On Furstenberg systems for some aperiodic multiplicative functions
The Chowla conjecture from 1965 predicts that all autocorrelations of the Liouville function vanish. In fact, after an adaptation, the Chowla conjecture was expected to hold for all aperiodic multiplicative functions with values in the unit disc (cf. Elliott’s conjecture from the 1990s). But in 2015, Matom\"aki, Radziwi\l\l and Tao gave a counterexample to Elliott’s conjecture by constructing aperiodic multiplicative functions (bounded by 1) for which (already) the Chowla conjecture of order 2 fails. During the talk I will explain however that the Chowla conjecture does not disappear entirely, in fact, for Matomäki, Radziwiłł, Tao’s functions it holds along a subsequence. The result is achieved by detecting certain dynamical systems (of algebraic origin),
called Furstenberg systems, obtained by examining autocorrelations along subsequences. Moreover, the same technique disproves a recent conjecture by Frantzikinakis and Host concerning logarithmic autocorrelations of multiplicative functions. The talk is based on my joint work with Alex Gomilko and Thierry de la Rue.