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

Abstract

Let $\alpha>0$ and $0<\gamma<1$. Define $g_{\alpha,\gamma}:\mathbb{N}\rightarrow\mathbb{N}$ 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 $IP^{\star}$-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{HSD}=\left\{ A\subseteq\mathbb{N}:\sum_{n\in A}\frac{1}{n}=\infty\right\} $ as this family is directly connected with the famous Erd\H{o}s sum of reciprocal conjecture and as a consequence we get $g_{\alpha,\gamma}\left[\mathbb{P}\right]\in\mathcal{HSD}$, 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-\v{C}ech compactifications of discrete semigroups.