簡易檢索 / 詳目顯示
| 研究生： |
鄭宇翔 Yu-Shiang Cheng |
|---|---|
| 論文名稱： |
加乘法估計在實數體中一些變化的探討 On a variant of sum-product estimate |
| 指導教授： |
沈俊嚴
Chun-Yen Shen |
| 口試委員: | |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2014 |
| 畢業學年度： | 102 |
| 語文別： | 中文 |
| 論文頁數： | 13 |
| 中文關鍵詞： | 加乘法估計 |
| 外文關鍵詞： | sum-product |
| 相關次數： | 點閱：7 下載：0 |
| 分享至: |
| 查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報 |
著名的 Erdös-Szemerédi 加乘法估計定理說明了任意的有限整數子集合的加法集或是乘法集的個數會有擴張現象。在這篇論文裡，我們探討一些此類估計的變形。我們證明了給定實數裡的任意一個有限子集，它的加法集或是平方加法集也會有擴張現象。我們證明的工具包含了離散幾何的一些定理以及加法組合學的一些技巧。
The well-known sum-product estimate of Erdös and Szemerédi asserts that any finite set in integers either has sum set or product set much larger than itself. In this thesis, we also study a variant of sum-product estimate in reals. We prove that for any finite set in reals either its sum set or sum of squares set is large. Our tools include some theorems from incidences geometry and additive combinatorics.
[1]{G.Elekes},{On the number of sums and products}, Acta Arith. 81 (1997), 265-367.
[2]{G. Elekes and I. Ruzsa},{Few sums, many products}, Studia Sci. Math. Hungar. 40 (2003) (3), 301-308.
[3]{G. Elekes and M. Nathanson},{Convexity and sumsets}, J. Number Theory 83 (2000), no. 2, 194–201.
[4]{I.Ruzsa},{An analog of Freiman's theorem in groups}, In Structure Theory of Set Addition, Ast\'{e}risque 258 (1999), 323-326.
[5]{J. Bourgain},{Mordell's exponential sum estimate revisited}, J. Amer. Math. Soc. 18 (2005) (2), 477–4993.
[6]{J. Bourgain and M-C chang}{Onthe size of K-fold sum and product sets of integers}, J.AmermMath. soc.17 (2004), no.2, 473-497.
[7]{J. Solymosi},{Bounding multiplicative energy by the sumset}, Adv. Math 222 (2009), 402-408.
[8]{J. Solymosi},{On the number of sums and products}, Bull. London Math. Soc. 37 (2005) (4), 491-494.
[9]{K. Ford},{Sums and products from a finite set of real numbers}, Ramanujan journal 2 (1998)(1-2), 59-66.
[10]{Liangpan Li and Oliver Roche-Newton},{Convexty and a sum-product type estimate}, arXiv:1111.5159.
[11]{L. Sz\'{e}kely},{Crossing numbers and hard Erd\"{o}s problems in discrete geometry}, Combin.
Probab. Comput. 6 (1997), 353–358.
[12]{M. Nathanson},{On sums and products of integers}, Proc.Am. Math. Soc. 125 (1997),9-16.
[13]{N. Katz and T. Tao},{Some connections between the Falconer and Furstenburg conjectures}, New York J. Math. 7 (2001), 148–187.
[14]{P.Erd\"{o}s, E.Szemer\'{e}di},{Onsums and products of integers}, Studies in Pure Mathematics,
Birkh\"{a}user, Basel, (1983), 203-208.
