Randomness in number theory

• Project/Area Number: 24340017
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kobe University
• Principal Investigator: Fukuyama Katusi 神戸大学, 理学(系)研究科(研究院), 教授 (60218956)
• Co-Investigator(Kenkyū-buntansha): NAKADA HITOSHI 慶應義塾大学, 理工学部, 教授 (40118980) ; SUGITA HIROSHI 大阪大学, 大学院理学研究科, 教授 (50192125) ; KAJINO NAOTAKA 神戸大学, 大学院理学研究科, 准教授 (90700352)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000); Fiscal Year 2014: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2013: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2012: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: 確率論 / 一様分布 / 差異量 / 等比数列 / 重複対数の法則 / uniform distribution / discrepancy / lacunary sequence / 経験分布過程 / 間隙級数 / 一様分布論 / 間隙列

Outline of Final Research Achievements

We investigated the uniform distributedness of geometric progression and made research on the law of the iterated logarithm which discribes the deviation from the uniform distribution. We tried to determine the constant and suceeded in having the general formula of the constant in the case the absolute value of the ratio is large.
It solves the problem on the speed of convergence to the uniform distribution when the ratio is large.

Research Products

• [Journal Article] The central limit theorem for complex Riesz-Raikov sums (2015)
  • Author(s): K. Fukuyama, N. Kuri
  • Journal Title: Comptes Rendus de l'Academie des Sciences, Paris, Serie I Volume: 351 Issue: 8 Pages: 749-753
  • DOI: 10.1016/j.crma.2015.04.020
  • NAID: 120005980724
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A metric discrepancy result for the sequence of powers of minus two (2014)
  • Author(s): Katusi Fukuyama
  • Journal Title: Indagationes Mathematicae Volume: 25 Issue: 3 Pages: 487-504
  • DOI: 10.1016/j.indag.2013.12.002
  • NAID: 120005980776
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Optimal bound for the discrepancies of lacunary sequences (2013)
  • Author(s): C. Aistleitner, K. Fukuyama, Y. Furuya
  • Journal Title: Acta Arithmetica Volume: 158 Issue: 3 Pages: 229-243
  • DOI: 10.4064/aa158-3-3
  • Peer Reviewed
• [Journal Article] Metric discrepancy results for alternating geometric progressions (2013)
  • Author(s): K. Fukuyama
  • Journal Title: Monatshefte fuer Mathematik Volume: 171 Issue: 1 Pages: 33-63
  • DOI: 10.1007/s00605-012-0419-4
  • NAID: 120005980782
  • Peer Reviewed
• [Journal Article] Bounded law of the iterated logarithm for discrepancies of permutations of lacunary sequences (2012)
  • Author(s): K. Fukuyama, Y. Mitsuhata
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B39 Pages: 65-88
  • Peer Reviewed
• [Journal Article] On the law of the iterated logarithm for trigonometric series with bounded gap (2012)
  • Author(s): C. Aistleitner, K. Fukuyama
  • Journal Title: Probability Theory and related Fields Volume: 154 Pages: 607-620
  • NAID: 120005980786
  • Peer Reviewed
• [Presentation] Metric discrepancy results for complex geometric porgressions (2015)
  • Author(s): 福山克司、久利典之
  • Organizer: 日本数学会統計数学分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] On permutational invariance of the metric discrepancy results (2015)
  • Author(s): 福山克司、野田裕太郎
  • Organizer: 日本数学会統計数学分科会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21
• [Presentation] 有界間隙列の重複対数の法則について ＩＩ (2014)
  • Author(s): 福山克司、Christoph Aistleitner
  • Organizer: 日本数学会統計数学分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25
• [Presentation] Metric discrepancy results for Erdos-Fortet sequence (2012)
  • Author(s): 福山克司、宮本翔
  • Organizer: 日本数学会秋季総合分科会統計数学分科会
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-18
• [Presentation] 等比数列の一様分布論
  • Author(s): 福山克司
  • Organizer: 日本数学会秋季総合分科会統計数学分科会特別講演
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] The central limit theorem for subsequences of Erdos-Fortet sequence
  • Author(s): 福山克司、簑原貴文
  • Organizer: 日本数学会年会統計数学分科会
  • Place of Presentation: 学習院大学