A dynamical approach to nonhomogeneous spectra

Let $\alpha \gt 0$ and $0 \lt \gamma \lt 1$. 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 $\mathbb N$: if a subset $A$ of $\mathbb N$ is an IP-set, a central set, an IP$^*$-set, or a central$^*$-set, then so is $g_{\alpha ,\gamma }(A)$ for all $\alpha \gt 0$ and $0 \lt \gamma \lt 1$. 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 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.