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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/92717完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 沈俊嚴 | zh_TW |
| dc.contributor.advisor | Chun-Yen Shen | en |
| dc.contributor.author | 江子緯 | zh_TW |
| dc.contributor.author | Tzu-Wei Chiang | en |
| dc.date.accessioned | 2024-06-13T16:11:16Z | - |
| dc.date.available | 2024-06-14 | - |
| dc.date.copyright | 2024-06-13 | - |
| dc.date.issued | 2024 | - |
| dc.date.submitted | 2024-06-11 | - |
| dc.identifier.citation | [Bou99] J. Bourgain. ”On triples in arithmetic progression”. Geom. Funct. Anal.,9(5):968–984, 1999.
[BS23] Bloom, Thomas F.; Sisask, Olof. ”An improvement to the Kelley-Meka bounds on three-term arithmetic progressions”.arXiv:2309.02353v1 [Rot53] K. F. Roth. ”On certain sets of integers”. J. London Math. Soc., 28:104–109,1953. [San11] Sanders, Tom. ”On Roth’s theorem on progressions”. Annals of Mathematics.,(2) 174 (2011), no. 1, 619-636. [SS07] Elias M. Stein,; Rami Shakarchi. ”Fourier analysis an introduction”. Princeton University Press, 2007. [TV06] T. C. Tao and H. V. Vu. ”Additive combinatorics”. Cambridge University Press, Cambridge, 2006. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/92717 | - |
| dc.description.abstract | Roth 定理是加性組合中的一個知名定理，此定理的敘述為：給定一個正整數子集合，且此集合不包含任何長度為 3 的等差數列，則我們能給予此集合大小的估計。在 Roth 定理之後，有許多數學家對此集合的大小估計進行改進，在此篇文章哩，我們將介紹 Bourgain 與 Sanders 對此集合大小估計的改進。 | zh_TW |
| dc.description.abstract | Roth’s theorem on arithmetic progressions is a result in additive combinatorics which states that if there is a set of positive integers that contains no non-trivial 3-term arithmetic progression, then we can give an estimate of the size of this set. There have been many refinements following Roth’s approach. In this paper, we will introduce the proofs of two refinements given by Bourgain and Sanders. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2024-06-13T16:11:15Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2024-06-13T16:11:16Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | 摘要 v
Abstract vii Contents ix Chapter 1 Introduction 1 Chapter 2 Bourgain’s Approach 3 2.1 Summary 3 2.2 Basic Definitions 5 2.3 Estimates on Bohr Sets 5 2.4 Regular Bohr Sets 9 2.5 Estimation of Exponential Sum 15 2.6 Estimate of density 21 2.7 Comparison of the Integrals 24 2.8 Density Increment (1) 29 2.9 Density Increment (2) 31 2.10 Conclusion 36 Chapter 3 Sanders’ Approach 41 3.1 Summary 41 3.2 Basic Definitions and Notations 42 3.3 Bohr Sets 44 3.4 Spectrum, Orthogonal and Dissociated 45 3.4.1 Spectrum and Orthogonal 45 3.4.2 Dissociated 48 3.5 Basic Properties of Bohr Sets 60 3.6 Bohr Sets as Majorants 65 3.7 Getting a Density Increment 76 3.8 Roth’s Theorem in High Rank Bohr Sets 88 3.9 The Main Result 93 Chapter 4 Concrete Proofs of Sanders’ Approach 109 4.1 Summary 109 4.2 Basic Definitions and Notations 110 4.3 Bohr Sets 112 4.4 Spectrum, Orthogonal and Dissociated 113 4.4.1 Spectrum and Orthogonal 113 4.4.2 Dissociated 117 4.5 Basic Properties of Bohr Sets 130 4.6 Bohr Sets as Majorants 132 4.7 Getting a Density Increment 145 4.8 Roth’s Theorem in High Rank Bohr Sets 154 4.9 The Main Result 155 References 159 | - |
| dc.language.iso | en | - |
| dc.subject | 哈代-李特爾伍德圓法 | zh_TW |
| dc.subject | Roth 定理 | zh_TW |
| dc.subject | Bohr 集合 | zh_TW |
| dc.subject | Salem-Spencer 集合 | zh_TW |
| dc.subject | 傅立葉分析 | zh_TW |
| dc.subject | Fourier Analysis | en |
| dc.subject | Salem-Spencer set | en |
| dc.subject | Bohr set | en |
| dc.subject | Circle Method | en |
| dc.subject | Roth’s Theorem | en |
| dc.title | 等差數列集合密度的研究與進展 | zh_TW |
| dc.title | Recent progress on the density of three term arithmetic progressions | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 112-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 林延輯;余韋亘 | zh_TW |
| dc.contributor.oralexamcommittee | Yan-Ji Lin;Wei-Hsuan Yu | en |
| dc.subject.keyword | Roth 定理,哈代-李特爾伍德圓法,傅立葉分析,Salem-Spencer 集合,Bohr 集合, | zh_TW |
| dc.subject.keyword | Roth’s Theorem,Circle Method,Fourier Analysis,Salem-Spencer set,Bohr set, | en |
| dc.relation.page | 159 | - |
| dc.identifier.doi | 10.6342/NTU202401107 | - |
| dc.rights.note | 同意授權(全球公開) | - |
| dc.date.accepted | 2024-06-11 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 數學系 | - |
| 顯示於系所單位： | 數學系 | |
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