Geometry of arithmetic qusicrystals

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K03504
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Meiji University
• Principal Investigator: Sunada Toshikazu 明治大学, 研究・知財戦略機構(中野), 研究推進員 (20022741)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: 離散集合 / 非周期的結晶構造 / アイゼンシュタイン数

Outline of Final Research Achievements

For more than two years, I was significantly ill due to an intractable disease (a type of vasculitis) and a kidney disease, and he was repeatedly hospitalized and discharged. Therefore, my research did not proceed as expected, and he could not go out due to the spread of the new corona. This is why I have not used a significant portion of his Grants-in-Aid for Scientific Research.
For this reason, unfortunately, I have not yet provided an overview of our achievements. However, studies of discrete sets in Euclidean space have yielded some results. The most interesting subject is the discrete set defined "arithmetically". The discrete set related to the Eisenstein number as an typical example has been the subject of research so far. I have obtained knowledge about the distribution from the viewpoint of generalized Riemann sum. It is also interesting as an example of aperiodic crystal composition. If I feel better, I will continue my research in this direction.

Free Research Field

Geometry

Academic Significance and Societal Importance of the Research Achievements

純粋数学のみならず、結晶構造の理論を通して物質科学の分野にも大きな意義を有している。病気のため、完成には至らなかったが、拙著の増補版を通して、広く数理科学の発展に貢献している。