Equivalent definitions of oscillating sequences of higher orders

TL;DR

This paper explores various equivalent ways to define oscillating sequences of higher orders by examining their disjointness from different dynamical systems on tori, enhancing understanding of their properties.

Contribution

It provides multiple equivalent characterizations of fully oscillating sequences in terms of their disjointness from diverse dynamical systems on tori.

Findings

Multiple equivalent definitions of oscillating sequences of all orders.

Connections established between oscillating sequences and dynamical systems on tori.

Framework for analyzing oscillating sequences through dynamical disjointness.

Abstract

An oscillating sequence of order $d$ is defined by the linearly disjointness from all $\{e^{2\pi i P(n)} \}_{n=1}^{\infty}$ for all real polynomials $P$ of degree smaller or equal to $d$. A fully oscillating sequence is defined to be an oscillating sequence of all orders. In this paper, we give several equivalent definitions of such sequences in terms of their disjointness from different dynamical systems on tori.