数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (1): 189-194.doi: 10.1016/S0252-9602(14)60150-8

• 论文 • 上一篇    下一篇

A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION

刘佳|孙钰   

  1. Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics,  |engbu 233030, China Faculty of Science, Jiangsu University, Zhenjiang 212013, China
  • 收稿日期:2013-12-04 修回日期:2014-01-08 出版日期:2015-01-20 发布日期:2015-01-20
  • 通讯作者: 孙钰,sunyu88sy@163.com E-mail:liujia860319@163.com;sunyu88sy@163.com
  • 基金资助:

    This work was supported by NSFC (11171124). †Corresponding author: Yu SUN.

A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION

LIU Jia,SUN Yu   

  1. Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics,  |engbu 233030, China Faculty of Science, Jiangsu University, Zhenjiang 212013, China
  • Received:2013-12-04 Revised:2014-01-08 Online:2015-01-20 Published:2015-01-20
  • Contact: SUN Yu,sunyu88sy@163.com E-mail:liujia860319@163.com;sunyu88sy@163.com
  • Supported by:

    This work was supported by NSFC (11171124). †Corresponding author: Yu SUN.

摘要:

Let Q be an infinite set of positive integers,  > 1 be a real number and let  (Q) = x 2 R :
x − q q− for infinitely many (p, q) 2 Z × Q. For any given positive integer m, set Q(m) = {n 2 N : (n,m) = 1}. If m is divisible by at least two prime factors, Adiceam [1] showed that W (N) \ W (Q(m)) contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers? In this note, we give an affirmative answer to Adiceam’s question.

关键词: liminf set, Diophantine approximation, Liouville number

Abstract:

Let Q be an infinite set of positive integers,  > 1 be a real number and let  (Q) = x 2 R :
x − q q− for infinitely many (p, q) 2 Z × Q. For any given positive integer m, set Q(m) = {n 2 N : (n,m) = 1}. If m is divisible by at least two prime factors, Adiceam [1] showed that W (N) \ W (Q(m)) contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers? In this note, we give an affirmative answer to Adiceam’s question.

Key words: liminf set, Diophantine approximation, Liouville number

中图分类号: 

  • 11J82