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analysis

Source: miklos schweitzer 1994 q4

October 16, 2021
real analysisirrational number

Problem Statement

For a given irrational number α\alpha , y1,α=αy_{1,\alpha} = \alpha. If yn−1,αy_{n-1, \alpha} is given, let yn,αy_{n, \alpha} be the first member of the sequence ({kα})k=1∞\big (\{k \alpha \} \big) ^ \infty_{k = 1} to fall in the interval (0,yn−1,α)(0, y_{n-1,\alpha}) ({ x } denotes the fraction of the number x ). Show that there exists an open set G⊂(0,1)G\subset (0,1) , which has a limit point 0 and for all irrational α\alpha , infinitely many members of the (yn,α)(y_{n,\alpha}) sequence do not belong to G.
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