On the multiple recurrence of infinite measure preserving transformations and a conjecture by Erdos

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K13766
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Keio University
• Principal Investigator: Nakada Hitoshi 慶應義塾大学, 理工学部(矢上), 名誉教授 (40118980)
• Research Collaborator: Aaronson Jon Tel Aviv University, Faculty of Exact Sciences, Professor ; Sarig Omri Weizmann Institute of Science, Faculty of Mathematics and Computer Science, Professor
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: エルゴード理論 / 無限大不変測度 / 等差数列のエルデシ予想

Outline of Final Research Achievements

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.

Free Research Field

数理解析学

Academic Significance and Societal Importance of the Research Achievements

等差数列に関するエルデシ予想は21世紀に入り、Green-Taoにより素数列に関しては解決されたものの、本来の問題は依然として未解決の難問である。本課題では、この問題解決への一つのアプローチとして1970年代に H. Furstenberg により提案された方法の厳密な正しさを証明することを意識しながら infinite ergodic theory を研究した。infinite ergodic theory の多重再帰性に関する研究の進展はエルデシ予想の解決に向けた一つの大きな可能性を持つもので、そこに本研究の学術的意義が見いだされる。