A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION  

作　　者：刘佳 孙钰 

机构地区：[1]Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics [2]Faculty of Science, Jiangsu University

出　　处：《Acta Mathematica Scientia》2015年第1期189-194,共6页数学物理学报（B辑英文版）

基　　金：supported by NSFC(11171124)

摘　　要：Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ（N） ／ Wτ（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.Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ（N） ／ Wτ（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 

分 类 号：O171[理学—数学]