Good Points for Diophantine Approximation

Proceedings Mathematical Sciences

Daniel Berend¹  ArtÅ«ras Dubickas² 

[1] Departments of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 0, Israel$$;Department of Mathematics and Informatics, Vilnius University, Vilnius LT-0, Lithuania$$

关键词: Uniform distribution;    diophantine approximation;    Hausdorff dimension.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Given a sequence $(x_n)^∞_{n=1}$ of real numbers in the interval [0,1) and a sequence $(𝛿_n)^∞_{n=1}$ of positive numbers tending to zero, we consider the size of the set of numbers in [0,1] which can be `well approximated’ by terms of the first sequence, namely, those $yin[0,1]$ for which the inequality $|y-x_n| < 𝛿_n$ holds for infinitely many positive integers 𝑛. We show that the set of `well approximable’ points by a sequence $(x_n)^∞_{n=1}$, which is dense in [0,1], is `quite large’ no matter how fast the sequence $(𝛿_n)^∞_{n=1}$ converges to zero. On the other hand, for any sequence of positive numbers $(𝛿_n)^∞_{n=1}$ tending to zero, there is a well distributed sequence $(x_n)^∞_{n=1}$ in the interval [0,1] such that the set of `well approximable’ points 𝑦 is `quite small’.

【 授权许可】

Unknown