A new development of irrational number theory

对于满足一系列线性Mahler函数方程的无限乘积函数，我们建立了其无理测度为有限的一个有用的充分条件。应用这一条件给出了无穷乘积函数值在整数倒数处代数无关性的一个新判据。利用消元理论证明了两个无穷乘积函数值在一般代数数上的代数无关性。此外，我们还研究了该方法的一种公理方法，它除了具有无穷多个乘积函数外，还具有一定的应用价值。

For infinite product functions satisfying a chain of linear Mahler functional equations we established a useful sufficient condition for finiteness of the irrationality measure of the infinite product functions. We applied the condition to show a new criterion for algebraic independence of the values of the infinite product functions at the reciprocal of an integer. We also proved algebraic independence of the values of two infinite product functions at a general algebraic number by using elimination theory. Moreover, we investigate an axiomatic way of our method, which would have certain applications besides the infinite product functions.