Oscillating sequences, Gowers norms and Sarnak's conjecture

TL;DR

This paper explores the properties of oscillating sequences of higher order, demonstrating their orthogonality to certain dynamical systems and nilsequences, while also providing examples with large Gowers norms and new estimates for the Möbius function.

Contribution

It establishes orthogonality of higher order oscillating sequences to specific dynamical systems and nilsequences, and introduces new bounds for the Möbius function using Bourgain's method.

Findings

Higher order oscillating sequences are orthogonal to $d$-nilsequences and systems with quasi-discrete spectrum.

Existence of oscillating sequences with large Gowers norms.

New estimates for the average of the Möbius function on short intervals.

Abstract

It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that any oscillating sequence of order $d$ is orthogonal to any $d$-nilsequence arising from the skew product on the $d$-dimensional torus $\mathbb{T}^d$. The proof yields that any oscillating sequence of higher order is orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum. however, we provide an example of oscillating sequence of higher order with large Gowers norms. We further obtain a new estimation of the average of M\"{o}bius function on the short interval by appealing to Bourgain's double recurence argument.