Hard Problem on Limit Points.

Asked Nov 13 '17 at 18:33

Active Nov 13 '17 at 18:37

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Find the set of all limit points of the sequence $a_{n}=n\alpha-\lceil n\alpha \rceil,\alpha\not \in \mathbb{Q}.$

This problem is particularly easy when $a\in \mathbb{Q}$, but somehow the same techniques fail to work when $\alpha$ is irrational. Any help/hint on this problem will be much appreciated.

edited Nov 13 '17 at 18:37

Alex Provost

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asked Nov 13 '17 at 18:33

Student

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Is that the floor or ceiling function? – Alex Provost Nov 13 '17 at 18:35
In my book it means the least integer greater than or equal to $n\alpha.$ – Student Nov 13 '17 at 18:36
I am not sure what the terms floor or ceiling mean.... – Student Nov 13 '17 at 18:36
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Then it's ceiling. See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions – Alex Provost Nov 13 '17 at 18:37
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When $\alpha$ is irrational, the set ${n\alpha - \lceil n\alpha \rceil : n\in\mathbb{N} }$ will be dense in the interval $[-1,0]$. Thus the set of all limit points is that interval. This is related. – Xander Henderson Nov 13 '17 at 18:39

Hard Problem on Limit Points.

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