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}. ...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.展开更多
文摘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.
