A dynamical approach to nonhomogeneous spectra

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 $g_{\alpha ,\gamma }(\mathbb{N})=\{g_{\alpha ,\gamma }(n)\text{\hspace{0.17em}⁣}:n\in \mathbb{N}\}$ 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${}^{\ast }$-set, or a central${}^{\ast }$-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.