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
In this paper, we develop an algebraic independence method originated from a work of Mordoukhay-Boltovskoy, which allows us to establish the algebraic independence of certain numbers connected with Liouville numbers. ...In this paper, we develop an algebraic independence method originated from a work of Mordoukhay-Boltovskoy, which allows us to establish the algebraic independence of certain numbers connected with Liouville numbers. This method depends either on elementary results of the Kummer theory or on transcendence measures for certain classes of numbers.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.
文摘In this paper, we develop an algebraic independence method originated from a work of Mordoukhay-Boltovskoy, which allows us to establish the algebraic independence of certain numbers connected with Liouville numbers. This method depends either on elementary results of the Kummer theory or on transcendence measures for certain classes of numbers.
