Metric studies on uniform distribution theory

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05204
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: Fukuyama Katusi 神戸大学, 理学研究科, 教授 (60218956)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 一様分布論 / 間隙級数論 / 確率論

Outline of Final Research Achievements

For given sequence of real numbers, multiplying a real number and investigate the distribution of fractional part. This is the thema of uniform distribution theory.
For various sequences, the limit distribution is the uniform distribution for almost every multiplied real number, and we analyse the speed of convergence toward the uniform distribution by using the notion of discrepancies. We have already prove the law of the iterated logarithm for discrepancies for geometric progressions, and proved the formula which give the limsup constant when the ratio is large. In this project, we found several ratios for which the limsup constant is different from the value given by the formula. We also proved the existence of the sequence which realize the given speed of convergence of discrepancies.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

解析的整数論の一分野として研究されてきた一様分布論に強力に確率論の手法を導入し、重複大数の法則を証明することにより、極めて初等的な等比数列について、今まで未知であった小数部分の分布の一様分布への漸近の速度について研究を進めている。公比が大きい場合にはすでに決定的な結果を得ており、この研究プロジェクトで小さい公比の場合に起こる様々なパターンの挙動例を見出すことに成功しており、新奇な実例を挙げたこととなっている。