A theorem on the transcendency and its applications (Q1373837)

This paper consists of three basic theorems. The first one describes that if the number \(\beta\) is good approximable then the degree of \(\beta\) is sufficiently large. The second theorem proves that if \(f(x)= [a_1x, a_2x,\dots]\) is a continued fraction function and \(\alpha\) is an algebraic number, then \(f(\alpha)\) is transcendental or algebraic with a sufficiently large degree. The third one describes that a certain product is transcendental or algebraic with a large degree too.