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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Piecewise syndetic set ： ウィキペディア英語版 Piecewise syndetic set In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set $S\; \backslash sub\; \backslash mathbb$ is called ''piecewise syndetic'' if there exists a finite subset ''G'' of $\backslash mathbb$ such that for every finite subset ''F'' of $\backslash mathbb$ there exists an $x\; \backslash in\; \backslash mathbb$ such that :$x+F\; \backslash subset\; \backslash bigcup\_\; (S-n)$ where $S-n\; =\; \backslash$. Equivalently, ''S'' is piecewise syndetic if there are arbitrarily long intervals of $\backslash mathbb$ where the gaps in ''S'' are bounded by some constant ''b''. == Properties == * A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set. * If ''S'' is piecewise syndetic then ''S'' contains arbitrarily long arithmetic progressions. * A set ''S'' is piecewise syndetic if and only if there exists some ultrafilter ''U'' which contains ''S'' and ''U'' is in the smallest two-sided ideal of $\backslash beta\; \backslash mathbb$, the Stone–Čech compactification of the natural numbers. * Partition regularity: if $S$ is piecewise syndetic and $S\; =\; C\_1\; \backslash cup\; C\_2\; \backslash cup\; ...\; \backslash cup\; C\_n$, then for some $i\; \backslash leq\; n$, $C\_i$ contains a piecewise syndetic set. (Brown, 1968) * If ''A'' and ''B'' are subsets of $\backslash mathbb$, and ''A'' and ''B'' have positive upper Banach density, then $A+B=\backslash$ is piecewise syndetic〔R. Jin, (Nonstandard Methods For Upper Banach Density Problems ), ''Journal of Number Theory'' 91, (2001), 20-38.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Piecewise syndetic set」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.