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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/92718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 | 王健安 | zh_TW |
| dc.contributor.author | Jian-An Wang | en |
| dc.date.accessioned | 2024-06-13T16:11:38Z | - |
| 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 | [1] T. Bloom and T. G. F. Jones. A sum-product theorem in function fields, 2013.
[2] J. Bourgain. More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory, 1(01):1–32, 2005. [3] J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, and applications. Geometric & Functional Analysis GAFA, 14(1):27–57, 2004. [4] G. Elekes. On the number of sums and products. Acta Arithmetica, 81(4):365–367, 1997. [5] P. Erdös. On sets of distances of n points. The American Mathematical Monthly, 53(5):248–250, 1946. [6] M. Z. Garaev. An explicit sum-product estimate in Fp, 2007. [7] M. Z. Garaev and C.-Y. Shen. On the size of the set a(a+1), 2008. [8] L.GuthandN.H.Katz. On the erdős distinct distances problem in the plane.Annals of mathematics, pages 155–190, 2015. [9] T. G. F. Jones. New quantitative estimates on the incidence geometry and growth of finite sets, 2013. [10] B.Murphy,G.Petridis,T.Pham,M.Rudnev,andS.Stevens.Onthepinneddistances problem in positive characteristic. Journal of the London Mathematical Society, 105(1):469–499, 2022. [11] G.Petridis.New proofs of plünnecke-type estimates for product sets in groups,2011. [12] C.-Y.Shen.Onthesumproductestimatesandtwovariablesexpanders.Publicacions Matemàtiques, pages 149–157, 2010. [13] J. Solymosi. On sum-sets and product-sets of complex numbers. Journal de théorie des nombres de Bordeaux, 17(3):921–924, 2005. [14] A. V. Sutherland. 9 local fields and hensel’s lemmas. 2019. [15] E. Szemerédi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3:381–392, 1983. [16] T. Tao and V. H. Vu. Additive combinatorics, volume 105. Cambridge University Press, 2006. [17] C. D. Tóth. The szemerédi-trotter theorem in the complex plane. Combinatorica, 35(1):95–126, feb 2015. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/92718 | - |
| dc.description.abstract | 對於任意無有限子環的體，我們首先建立一個精確的 Bourgain-Katz-Tao 點線 重合上界。再者，在非阿基米德局部體上，我們給出一個強加乘集界。最後，結合前面兩種結果，我們證明一個非阿基米德局部體平面上的點線重合上界。
除此之外，應用我們證明的點線重合界，能夠探討非阿基米德局部體上的相異距離問題與擴展者問題。 | zh_TW |
| dc.description.abstract | First, we establish an explicit upper bound for the Bourgain-Katz-Tao’s point-line incidence theorem over fields without any finite subrings. Second, we obtain a stronger sum-product bound over non-archimedean local fields. Furthermore, by combining the previous two results, we prove an upper bound for point-line incidence over non-archimedean local fields.
As an application, we use our incidence bounds to study the distinct distance problem and the expander problems over non-archimedean local fields. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2024-06-13T16:11:38Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2024-06-13T16:11:38Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | 摘要 (Page iii)
Abstract (Page v) Contents (Page vii) Chapter 1 Introduction (Page 1) 1.1 Background............................... 1 Chapter 2 Main Results (Page 3) 2.1 First Result............................... 3 2.1.1 Main Tools .............................. 4 2.1.2 Main Lemmas............................. 5 2.1.3 Proof of the First Result........................ 7 2.2 Second Result.............................. 15 2.2.1 Main Tools .............................. 16 2.2.2 Main Lemmas............................. 19 2.2.3 Proof of the Second Result ............................. 28 2.3 Main Result............................... 29 Chapter 3 Applications (Page 31) 3.1 Distinct Distance Problem ....................... 31 3.1.1 Isotropic Vectors ........................... 32 3.1.2 A Lower Bound on the Pinned Distance . . . . . . . . . . . . . . . 35 3.2 Expanders Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Via the Point-Line Incidence ..................... 41 3.2.2 Via the Stronger Sum-Product Theorem . . . . . . . . . . . . . . . 47 References (Page 49) | - |
| dc.language.iso | en | - |
| dc.subject | 加乘集界 | zh_TW |
| dc.subject | 點線重合 | zh_TW |
| dc.subject | 局部體 | zh_TW |
| dc.subject | 非阿基米德 | zh_TW |
| dc.subject | Non-Archimedean | en |
| dc.subject | Local Fields | en |
| dc.subject | Sum-Product Bound | en |
| dc.subject | Point-Line Incidence | en |
| dc.title | 非阿基米德局部體上的點線重合界 | zh_TW |
| dc.title | A Point-Line Incidence Bound over Non-Archimedean Local Fields | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 112-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 林延輯;俞韋亘 | zh_TW |
| dc.contributor.oralexamcommittee | Yen-chi Roger Lin;Wei-Hsuan Yu | en |
| dc.subject.keyword | 點線重合,加乘集界,非阿基米德,局部體, | zh_TW |
| dc.subject.keyword | Point-Line Incidence,Sum-Product Bound,Non-Archimedean,Local Fields, | en |
| dc.relation.page | 50 | - |
| dc.identifier.doi | 10.6342/NTU202401132 | - |
| dc.rights.note | 未授權 | - |
| dc.date.accepted | 2024-06-12 | - |
| 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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