A dynamical approach to nonhomogeneous spectra

TL;DR

This paper investigates the preservation of largeness properties of subsets of natural numbers under certain spectral functions, using a dynamical systems approach to unify and extend previous results.

Contribution

It introduces a dynamical framework to analyze how spectral functions preserve various largeness notions, providing unified proofs and new insights.

Findings

Spectral functions preserve largeness properties like IP-sets and central sets.

A dynamical correspondence links spectral preservation to suspension lift properties.

New results on the preservation of other largeness notions are obtained.

Abstract

Let $\alpha>0$ and $0<\gamma<1$. Define $g_{\alpha,\gamma}\colon \mathbb{N}\to\mathbb{N}_0$ by $g_{\alpha,\gamma}(n)=\lfloor n\alpha +\gamma\rfloor$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$. The set $g_{\alpha,\gamma}(\mathbb{N})=\{g_{\alpha,\gamma}(n)\colon n\in\mathbb{N}\}$ is called the $\gamma$-nonhomogeneous spectrum of $\alpha$. By extension, the functions $g_{\alpha,\gamma}$ are referred to as spectra. In 1996, Bergelson, Hindman and Kra showed that the functions $g_{\alpha,\gamma}$ preserve some largeness of subsets of $\mathbb{N}$, that is, if a subset $A$ of $\mathbb{N}$ is an IP-set, a central set, an IP$^*$-set, or a central$^*$-set, then $g_{\alpha,\gamma}(A)$ is the corresponding object for all $\alpha>0$ and $0<\gamma<1$. In 2012, Hindman and Johnson extended this result to include several other notions of largeness: C-sets, J-sets, strongly central sets, and piecewise syndetic sets. We adopt a dynamical approach to this issue and build a correspondence between the preservation of spectra and the lift property of suspension. As an application, we give a unified proof of some known results and also obtain some new results.