Cugiani-Mahler定理的推广

IF 1.2 2区 数学 Q1 MATHEMATICS Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze Pub Date : 2007-01-01 DOI:10.14288/1.0044613
Y. Bugeaud
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引用次数: 6

摘要

1955年,Roth证明了如果ξ是一个无理数,使得有一个正实数e和无穷多个有理数p/q且q≥1和|ξ−p/q| < q−2−e,则ξ是超越的。几年后,Cugiani用一个函数q→e(q)代替了e,得到了同样的结论,该函数非常缓慢地减小到零,前提是|ξ−p/q| < q−2−e(q)的有理解序列在适当意义上足够密集。我们给出了另一种更简单的库吉亚尼定理的证明,并将其推广到同时逼近。数学学科分类(2000):11J68。
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Extensions of the Cugiani-Mahler theorem
In 1955, Roth established that if ξ is an irrational number such that there are a positive real number e and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q−2−e , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with e replaced by a function q → e(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q−2−e(q) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous approximation. Mathematics Subject Classification (2000): 11J68.
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
90
审稿时长
>12 weeks
期刊介绍: The Annals of the Normale Superiore di Pisa, Science Class, publishes papers that contribute to the development of Mathematics both from the theoretical and the applied point of view. Research papers or papers of expository type are considered for publication. The Annals of the Normale Scuola di Pisa - Science Class is published quarterly Soft cover, 17x24
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