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KAKEN_16K13766seika.pdf
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Download
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無限大不変測度を持つエルゴード的変換の多重再帰性とエルデシ予想
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ムゲンダイ フヘン ソクド オ モツ エルゴードテキ ヘンカン ノ タジュウ サイキセイ ト エルデシ ヨソウ
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Mugendai fuhen sokudo o motsu erugōdoteki henkan no tajū saikisei to erudeshi yosō
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On the multiple recurrence of infinite measure preserving transformations and a conjecture by Erdos
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仲田, 均
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ナカダ, ヒトシ
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Nakada, Hitoshi
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慶應義塾大学・理工学部 (矢上)・名誉教授
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Research team head
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科研費研究者番号 : 40118980
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Aaronson, Jon
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Tel Aviv University・Faculty of Exact Sciences・Professor
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Collaborator
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Sarig, Omri
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Weizmann Institute of Science・Faculty of Mathematics and Computer Science・Professor
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Collaborator
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2019
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科学研究費補助金研究成果報告書
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2018
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自然数の部分列に対してその逆数の和が発散するとき、その中に任意の長さの等差数列が存在するというエルデシの未解決予想の解決を目指して、エルゴード理論からどのようにアプローチできるかを研究した。この問題のためには、古典的なエルゴード的変換の多重再帰性ではなく無限大の不変測度を持つエルゴード的変換の多重再帰性を考える必要がある。本研究ではこの観点から無限大不変測度を持つエルゴード的変換の研究を行い、Rauzy inductionの自然拡大、cylinder flowに関する極限定理について新しい結果を得た。更に、多重再帰性について、特別な性質を持つエルゴード的変換の存在を示すことに成功した。
We consider the following the long standing open question which is called the Erdos conjecture for arithmetic progressions : Suppose that (a_n) is a sub-sequence of natural numbers such that the sum of the inverse 1/a_n diverges. Then for any natural number k, there exists an arithmetic progression of length k in (a_n). The aim of this research is to find a way to solve this conjecture from infinite ergodic theory. In this point of vie w, we got the following results. (1) We constructed the natural extension of the Rauzy induction as a map on the set of translation surfaces. (2) We have some limit theorems for cylinder flows. Moreover we constructed an infinite measure preserving transformations which has a restricted mutiplicity of recurrence.
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研究種目 : 挑戦的萌芽研究
研究期間 : 2016～2018
課題番号 : 16K13766
研究分野 : 数理解析学
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