Fully oscillating sequences and weighted multiple ergodic limit

Abstract

We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems.

On uniformity of q‐multiplicative sequences

“…The interest of Theorem is that for ‐multiplicative sequences, the Gelfond property of order 1 implies automatically the Gelfond property of higher orders. As proved in , once we prove the Gelfond property , we immediately get a weighted ergodic theorem with weights along polynomials (see Theorem ).…”

“…For many ‐automatic sequences, we have a much stronger estimate for a constant . Using the terminology introduced in , means that is of Gelfond type with Gelfond exponent bounded by ; this condition can be construed as a weak notion of uniformity. Examples of ‐multiplicative sequences which are of Gelfond type include the Thue–Morse sequence and more generally all non‐periodic strongly ‐multiplicative sequences (see Proposition for details).…”

“…Following , we also introduce a property stronger than being an oscillating sequence. We shall say that is a sequence of Gelfond type of order if there exists such that Note that can only hold for .…”

Topological Wiener-Wintner ergodic theorem with polynomial weights

“…One motivation of the present work is to find topological dynamical systems (X, T ) and continuous functions f such that (f (T n x)) is fully oscillating or oscillating of order d for all x ∈ X without exception. If T is an affine dynamics of zero entropy on a compact abelian group, there is no such function different from zero which gives fully oscillating sequences [21,63]. But as we shall see, we can find such functions for some nilsystems, like ergodic nilsystems on Heisenberg homogeneous spaces.…”

“…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] .…”