Essential $\mathcal{F}$-sets of $\mathbb{N}$ under nonhomogeneous spectra

Abstract

Let $\alpha >0$ and $0<\gamma <1$. Define $g_{\alpha ,\gamma }:\mathbb{N}\to \mathbb{N}$ by $g_{\alpha ,\gamma }\left(n\right)=\lfloor \alpha n+\gamma \rfloor$. The set $\left\{g_{\alpha ,\gamma }\left(n\right):n\in \mathbb{N}\right\}$ is called the nonhomogeneous spectrum of $\alpha$ and $\gamma$. We refer to the maps $g_{\alpha ,\gamma }$ as nonhomogeneous spectra. In \cite{BHK}, Bergelson, Hindman and Kra showed that if $A$ is an $IP$-set, a central set, an $I{P}^{\star }$-set, or a central${}^{\star }$-set, then $g_{\alpha ,\gamma }\left[A\right]$ is the corresponding objects. Hindman and Johnsons extended this result to include several other notions of largeness: $C$-sets, $J$-sets, strongly central sets, piecwise syndetic sets, $AP$-sets syndetic set, $C^{\star }$-sets, strongly central${}^{\star }$- sets . In \cite{DHS}, De, Hindman and Strauss introduced $C$-set and $J$-set and showed that $C$-sets satisfy the conclusion of the Central Sets Theorem. To prepare this article, we have been strongly motivated by the fact that $C$-sets are essential $\mathcal{J}$-sets. In this article, we prove some new results regarding nonhomogeneous spectra of essential $\mathcal{F}$-sets for shift invariant $\mathcal{F}$-sets. We have a special interest in the family $\mathcal{H}\mathcal{S}\mathcal{D}=\left\{A\subseteq \mathbb{N}:{\sum }_{n\in A}\frac{1}{n}=\infty \right\}$ as this family is directly connected with the famous Erdős sum of reciprocal conjecture and as a consequence we get $g_{\alpha ,\gamma }\left[\mathbb{P}\right]\in \mathcal{H}\mathcal{S}\mathcal{D}$, where $\mathbb{P}$ is the set of prime numbers in $\mathbb{N}$. Throughout this article, we use some elementary techniques and algebra of the Stone-Čech compactifications of discrete semigroups.

15 pages. arXiv admin note: substantial text overlap with arXiv:2211.12372