Essential $\mathcal{F}$-sets of $\mathbb{N}$ under nonhomogeneous spectra
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| Natura: | Preprint |
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2022
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| _version_ | 1866910427702624256 |
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| author | Debnath, Pintu |
| author_facet | Debnath, Pintu |
| contents | Let $α>0$ and $0<γ<1$. Define $g_{α,γ}:\mathbb{N}\rightarrow\mathbb{N}$ by $g_{α,γ}\left(n\right)=\lfloorαn+γ\rfloor$. The set $\left\{ g_{α,γ}\left(n\right):n\in\mathbb{N}\right\} $ is called the nonhomogeneous spectrum of $α$ and $γ$. We refer to the maps $g_{α,γ}$ 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 $g_{α,γ}\left[A\right]$ 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 Erdős sum of reciprocal conjecture and as a consequence we get $g_{α,γ}\left[\mathbb{P}\right]\in\mathcal{HSD}$, 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-Čech compactifications of discrete semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_10951 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Essential $\mathcal{F}$-sets of $\mathbb{N}$ under nonhomogeneous spectra Debnath, Pintu Combinatorics 05D10, 22A15, 54D35 Let $α>0$ and $0<γ<1$. Define $g_{α,γ}:\mathbb{N}\rightarrow\mathbb{N}$ by $g_{α,γ}\left(n\right)=\lfloorαn+γ\rfloor$. The set $\left\{ g_{α,γ}\left(n\right):n\in\mathbb{N}\right\} $ is called the nonhomogeneous spectrum of $α$ and $γ$. We refer to the maps $g_{α,γ}$ 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 $g_{α,γ}\left[A\right]$ 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 Erdős sum of reciprocal conjecture and as a consequence we get $g_{α,γ}\left[\mathbb{P}\right]\in\mathcal{HSD}$, 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-Čech compactifications of discrete semigroups. |
| title | Essential $\mathcal{F}$-sets of $\mathbb{N}$ under nonhomogeneous spectra |
| topic | Combinatorics 05D10, 22A15, 54D35 |
| url | https://arxiv.org/abs/2212.10951 |