康托爾集之法瓦德長之極限行為

Limit Behavior of Favard Length of Cantor sets

吳悠(Yu Wu)

指導教授 ： 沈俊嚴

國立臺灣大學/理學院/數學系/碩士(2023年)

https://doi.org/10.6342/NTU202302276

摘要

在過去對於古典布豐投針的研究中，法瓦德長之概念被提出。法瓦德長是一種以集合對各個方向之投影來度量的幾何量。在R2中一個特定的集合，四角康托爾集，之法瓦德長已被研究數年。根據別西科維奇投影定理，四角康托爾集之法瓦德長為零。一個自然的問題是研究對於其康托爾方塊之法瓦德長極限行為之量化描述。本篇論文之目的在於調查過去關於四角康托爾集之法瓦德長問題，及討論將過去發展之方法推廣至五角康托爾集之可行性。

關鍵字

法瓦德長 ； 康托爾集 ； 別西科維奇投影 ； 幾何測度論

並列摘要

From the studies of classical Buffon needle problem, the concept of Favard length had been investigated. It is a geometric quantity of a set by measuring its projections behaviors on lines. In R2, a particular case, the four-corner Cantor set, has been studied for years. By Besicovitch projection theorem, the Favard length of four-corner Cantor set is zero. A nature question that was asked is to establish a quantitative rate of the convergence of Favard length in terms of its n-th generation. The purposes of this thesis are to survey the limit behavior of Favard length of Cantor set and Cantor-like set in R2 and discuss if the pioneer’s result can be generalized to five-corner Cantor set.

並列關鍵字

Favard length ； Cantor sets ； Besicovitch projection ； geometric-measure theory

參考文獻

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A.S. Besicovitch. Tangential properties of sets and arcs of infinite linear measure. Bulletin of the American Mathematical Society, 60(3):353-359, 1960.

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A. Volberg. Buffon needle: The probability of Buffon needle to land near Cantor set. EIMI Winter School, 2021.