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.
Categories: math.CO