Automorphisms with Quasi-discrete Spectrum, Multiplicative Functions and Average Orthogonality Along Short Intervals

Abstract

We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial P ∈ R[x] with irrational leading coefficient and for each multiplicative function ν : N → C, |ν| ≤ 1, we have 1 M M ≤m<2M 1 H m≤n<m+H e 2πiP (n) ν(n) −→ 0 as M → ∞, H → ∞, H/M → 0.

Measure complexity and Möbius disjointness

“…By the work of El Abdalaoui, Lemańcyzk and de la Rue [3], Möbius disjointness conjecture holds for any topological model of an ergodic system with irrational discrete spectrum (in fact in [3] Möbius disjointness conjecture is proved for any topological model of a totally ergodic system with quasi-discrete spectrum. We note that any ergodic automorphism with irrational discrete spectrum has quasi-discrete spectrum and totally ergodic).…”

“…It is known that by Green and Tao [18] that nilsystems satisfy the conjecture. We refer to [31,16,5,6,7,2,29,28,27,32,3,38,11,40,21,1,41,22] for the progress on this conjecture.…”

Fully oscillating sequences and weighted multiple ergodic limit

“…See [19], [20] for Sarnak's conjecture. See [1], [2], [6], [9], [14], [17], [15], [23] for some related recent works. The Möbius sequence (µ(n)) is a typical example of fully oscillating sequences ( [5], [13]).…”

Topological Wiener-Wintner ergodic theorem with polynomial weights

“…Namely, for a given sequence (w n ), we would like to find those topological dynamical systems (X, T ) of zero entropy such that for any f ∈ C(X) and any x ∈ X. Sarnak's conjecture states that the limit in (1.3) is zero for all systems of zero entropy when (w n ) is the Möbius function. Sarnak's conjecture is proved for different systems [10,11,12,14,17,18,19,20,21,23,25,30,31,41,43,42,56,58,66,64] .…”