本文選題：丟番圖逼近　切入點：質(zhì)量分布原理　出處：《華中科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：丟番圖逼近是數(shù)論中一個重要的分支,歷史悠久。近些年來,研究在這方面的取得了很大進(jìn)步。丟番圖逼近的內(nèi)容非常豐富,其主要內(nèi)容就是實數(shù)的有理逼近問題。眾所周知,實數(shù)的有理逼近問題與連分?jǐn)?shù)聯(lián)系密切,求一個數(shù)的連分?jǐn)?shù)展開式,可以轉(zhuǎn)換成構(gòu)造有理逼近解的問題,進(jìn)而連分?jǐn)?shù)成為了研究實數(shù)有理逼近的一個有力工具。本文在Adiceam、Jarn′?k和Besicovith工作的基礎(chǔ)上研究并改進(jìn)了可被有理數(shù)精確逼近的實數(shù)所構(gòu)成的下極限集的Hausdorff維數(shù)。記Wτ= Wτ(N) = {x ∈ R : |x- p/q| q-τ, i.o.(p, q) ∈ N2},Wτ(Q)={x∈R:|x-p/q|q-τi.o.(p,q)∈N×Q}.設(shè)Q?N(其中Q是無窮子集)�？紤]下極限集Wτ\Wτ(Q)的Hausdorf f維數(shù),本文證明了對任意的τ2,如果Q是一個N\Q-自由集,那么Wτ\Wτ(Q)的維數(shù)和Wτ的Hausdorff維數(shù)一樣,都是2/τ。另一方面,給出了一個當(dāng)τ≤2時,對一般的Q,dimHWτ\Wτ(Q)=0?=1+υ(N\Q)τ的例子。本文分為四部分:第一章為緒論,主要介紹所研究問題的背景及其意義,并簡述了國內(nèi)外關(guān)于此問題的研究現(xiàn)狀和相關(guān)的結(jié)論。第二章介紹了相關(guān)的預(yù)備知識,主要包括Hausdorff測度的定義及相關(guān)性質(zhì),連分?jǐn)?shù)及其一些性質(zhì),質(zhì)量分布原理及其他一些后文運(yùn)用到的結(jié)論。第三章給出了本文的主要結(jié)論及其證明過程,在證明過程中構(gòu)造Cantor集、測度以及長度度量的定義,根據(jù)以上的構(gòu)造,運(yùn)用質(zhì)量分布原理證明本文的結(jié)論。第四章對本文主要結(jié)論的補(bǔ)充及其證明。最后一章主要是探討相關(guān)結(jié)論的推廣問題。
[Abstract]:Diophantine approximation is an important branch of number theory and has a long history. In recent years, great progress has been made in the research of Diophantine approximation. The content of Diophantine approximation is very rich, and its main content is the rational approximation of real numbers. The problem of rational approximation of real numbers is closely related to continuous fractions. Finding a continuous fractional expansion of a number can be transformed into a problem of constructing rational approximation solutions, and then the continuous fraction becomes a powerful tool for studying rational approximation of real numbers. Based on the work of K and Besicovith, we study and improve the Hausdorff dimension of the lower limit set composed of real numbers which can be accurately approximated by rational numbers. Let W 蟿 = W 蟿 N) = {x 鈭，

本文編號：1623673