Images of \(C\) sets and related large sets under nonhomogeneous spectra


Let $\alpha > 0$ and $0 < \gamma < 1$. Define $ g_{\alpha, \gamma} \colon \mathbb{N} \to \mathbb{N}$ by $g_{\alpha, \gamma}(n) = \lfloor \alpha n + \gamma \rfloor$. The set $\{ g_{\alpha, \gamma}(n) {\,:\,} n \in \mathbb{N}\}$ is called the nonhomogeneous spectrum of $\alpha$ and $\gamma$. By extension, we refer to the maps $g_{\alpha, \gamma}$ as spectra. Bergelson, Hindman, and Kra showed that if $A$ is an $IP$-set, a central set, an $IP^*$-set, or a central$^*$-set, then $g_{\alpha, \gamma}[A]$ is the corresponding object. We extend this result to include several other notions of largeness: $C$-sets, $J$-sets, strongly central sets, and piecewise syndetic sets. Of these, $C$-sets are particularly interesting, because they are the sets which satisfy the conclusion of the central sets theorem (and so have many of the strong combinatorial properties of central sets) but have a much simpler elementary description than do central sets.

Integers (2012/ 13) 12B, 1–25