Yesterday Prof. Feng gave a seminar which was about Furstenberg’s x2, x3 theorem. In the talk, he used the following fact:
Let : , . Arrange in an increasing order . Then
He said the proof was easy and it was left as an exercise, with the hint that is an irrational number.
However, after I discussed this question with some undergraduates and postgraduates, we couldn’t figure out a rigorous proof in short time as there is no good pattern in this sequence. (Yuen Chi Ho suggested that one could approach this question by using Hurwitz’s theorem, but I don’t know how to use it.)
Can anyone solve this “easy” exercise?