傅氏分析在組合學的應用與Roth定理

Applications of finite Fourier analysis to combinations and Roth theorem

廖常至(Chung-Chih Liao)

指導教授 ： 沈俊嚴

國立中央大學/理學院/數學系/碩士(2015年)

摘要

Roth 定理闡述了如果一個正整數的子集的"密度"大於0,則它包含一個長度為3的等差數列。在本篇論文,我們探討了傅氏分析在組合學上的一些運用。除此之外,利用一些基本數論的結果,我們了解如何使用傅氏分析來證明Roth 定理。

關鍵字

Roth定理 ； 傅氏分析 ； 組合學

並列摘要

The celebrated result of Roth asserts that there exists an arithmetic progression of length three in a subset in integers with positive upper density. The result has been reproved and generalized later by many people. In this thesis, we study the approaches of Fourier analysis methods. We will see that the Finite Fourier analysis is powerful enough to prove the Roth theorem.

並列關鍵字

Roth ； Fourier analysis ； combinations

參考文獻

Heath-Brown, David Rodney. "Integer sets containing no arithmetic progressions." J. London Math. Soc.(2) 35.3 (1987): 385-394.

Iosevich, Alex. "Roth’s theorem on arithmetic progressions." (2003).

Szemerédi, Endre. "Integer sets containing no arithmetic progressions." Acta Mathematica Hungarica 56.1-2 (1990): 155-158.

Tao, Terence, and Van H. Vu. {it Additive combinatorics}. Vol. 105. Cambridge University Press, 2006, ch.4.