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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 Erdős–Szemerédi theorem ： ウィキペディア英語版 Erdős–Szemerédi theorem In arithmetic combinatorics, the Erdős–Szemerédi theorem, proven by Paul Erdős and Endre Szemerédi in 1983,〔.〕 states that, for every finite set of real numbers, either the pairwise sums or the pairwise products of the numbers in the set form a significantly larger set. More precisely, it asserts the existence of positive constants ''c'' and $\backslash varepsilon$ such that :$\backslash max(\; |A+A|,\; |A\; \backslash cdot\; A|\; )\; \backslash geq\; c\; |A|^$ whenever ''A'' is a finite non-empty set of real numbers of cardinality |''A''|, where $A+A\; =\; \backslash$ is the sum-set of ''A'' with itself, and $A\; \backslash cdot\; A\; =\; \backslash$. It is possible for ''A'' + ''A'' to be of comparable size to ''A'' if ''A'' is an arithmetic progression, and it is possible for ''A'' · ''A'' to be of comparable size to ''A'' if ''A'' is a geometric progression. The Erdős–Szemerédi theorem can thus be viewed as an assertion that it is not possible for a large set to behave like an arithmetic progression and as a geometric progression simultaneously. It can also be viewed as an assertion that the real line does not contain any set resembling a finite subring or finite subfield; it is the first example of what is now known as the ''sum-product phenomenon'', which is now known to hold in a wide variety of rings and fields, including finite fields.〔.〕 It was conjectured by Erdős and Szemerédi that one can take $\backslash varepsilon$ arbitrarily close to 1. The best result in this direction currently is by Solymosi,〔.〕 who showed that one can take $\backslash varepsilon$ arbitrarily close to 1/3. ==References== 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「Erdős–Szemerédi theorem」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.