论文标题
Essential $ \ Mathcal {f} $ - $ \ mathbb {n} $的集合下非均匀光谱
Essential $\mathcal{F}$-sets of $\mathbb{N}$ under nonhomogeneous spectra
论文作者
论文摘要
令$α> 0 $和$ 0 <γ<1 $。定义$ g_ {α，γ}：\ mathbb {n} \ rightarrow \ mathbb {n} $ by $ g_ {α，γ} \ left（n \ right）= \lfloorαn+γ\ rfloor $。集合$ \ left \ {g_ {α，γ} \ left（n \ right）：n \ in \ mathbb {n} \ right \} $称为$α$ $α$和$γ$的非均匀频谱。我们将地图$ g_ {α，γ} $称为非均匀光谱。在\ cite {bhk}中，Bergelson，Hindman和Kra表明，如果$ a $是$ ip $ - set，中央集合，$ ip^{\ star} $ - set或central $^{\ star} $ - set，则$ g_ {α，γ} \ weft [a \ weft [a \ right] $是$ g_ {al g_ {al g_ {\ g_ {\ g_ { Hindman和Johnsons扩大了此结果，其中包括其他几个宽敞的概念：$ c $ set，$ j $ - 集，强烈的中心集，PICEWIESSICTION SYNDETIC SETS，$ ap $ -sets syndetic Sets，$ c^{\ star} $ - sets，强烈的central $^{\ star} $ - 集合。在\ cite {dhs}，de，hindman和Strauss中引入了$ c $ - set和$ j $ - set，并表明$ c $ sets满足了中央集合定理的结论。为了准备这篇文章，我们一直以$ c $ set是必不可少的$ \ mathcal {j} $ sets-setss的事实的强烈动机。在本文中，我们证明了有关$ \ mathcal {f} $的非均匀光谱的一些新结果 - 用于换档不变$ \ mathcal {f} $ sets-sets的设置。 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} $，其中$ \ mathbb {p} $是$ \ mathbb {n} $中的一组素数。在本文中，我们使用了一些基本技术和代数的离散半群的Stone-chech压缩。
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.
