Probabilistic study on uniform distribution theory

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03518
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Kobe University
• Principal Investigator: Fukuyama Katusi 神戸大学, 理学研究科, 教授 (60218956)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 等比数列 / 一様分布論 / 差異量

Outline of Final Research Achievements

We studied the asymptotic behaviour of the discrepancies of geometric progression. If we perturb geometric progression with almost every initial term by using irrational rotation, the dependence as the stationary sequence seems to be eliminated by viewing the law of the iterated logarithm for discrepancies. We can also see the dependence not necessary vanishes by other perturbation, and see that any dependence smaller that original one can be realized by using some appropriately chosen perturbation. As to the subsequence of geometric progressoin, sometimes the dependence seems to be vanished, but we can take subsequence of it to make it recover the dependence of original sequence in the sense of the law of the iterated logarithm for discrepancies.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

等比数列の差異量の漸近挙動については判明していないことが大半であったが、測度論的手法を用いることによりほとんどすべての初期値に関して理論を展開することが可能になってきた。特に部分列の挙動や摂動の影響など等比数列から派生する様々な問題に関して知見が付け加わったことにより、一様分布論の測度論的研究の進展に寄与したものである。