elias001
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- I would like to know how to show ##S_{n}(\alpha)=\sum_{k=1}^{n} (-1)^{\lfloor{k\alpha\rfloor}}##.
For the following question, I am not sure how to go about solving it.
I tried the following computational examples to check if the statement is true.
##S_{n}(\alpha)## for:
##n = 1## and ##\alpha = \pi##, ##S_{1}=(\pi)=-1##
##n = 2## and ##\alpha = \pi##, ##S_{2}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} = (-1) + 1 = 0##
##n = 3## and ##\alpha = \pi##, ##S_{3}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} + (-1)^{\lfloor{3\pi\rfloor}} = (-1) + 1 + (-1) = -1##
##n = 4## and ##\alpha = \pi##, ##S_{4}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} + (-1)^{\lfloor{3\pi\rfloor}} + (-1)^{\lfloor{4\pi\rfloor}} = (-1) + 1 + (-1) + 1 = 0##
For:
##n = 1## and ##\alpha =e##, ##S_{1}=(e)=1##
##n = 2## and ##\alpha =e##, ##S_{2}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} = 1 + (-1) = 0##
##n = 3## and ##\alpha =e##, ##S_{3}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} + (-1)^{\lfloor{3e\rfloor}} = 1 + (-1) + 1 = 1 ##
##n = 4## and ##\alpha =e##, ##S_{4}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} + (-1)^{\lfloor{3e\rfloor}} + (-1)^{\lfloor{4e\rfloor}} = 1 + (-1) + 1 + 1 = 2##
Thank you in advance.
For ##n\in \mathbb{N}## and ##\alpha \in \mathbb{R}##, let ##S_{n}(\alpha)=\sum_{k=1}^{n} (-1)^{\lfloor{k\alpha\rfloor}}.## Prove that if ##\alpha## is irrational, then ##S_{n}(\alpha)=0## for infinitely many ##n \in \mathbb{N}##
I tried the following computational examples to check if the statement is true.
##S_{n}(\alpha)## for:
##n = 1## and ##\alpha = \pi##, ##S_{1}=(\pi)=-1##
##n = 2## and ##\alpha = \pi##, ##S_{2}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} = (-1) + 1 = 0##
##n = 3## and ##\alpha = \pi##, ##S_{3}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} + (-1)^{\lfloor{3\pi\rfloor}} = (-1) + 1 + (-1) = -1##
##n = 4## and ##\alpha = \pi##, ##S_{4}=(\pi)= (-1)^{\lfloor{\pi\rfloor}} +(-1)^{\lfloor{2\pi\rfloor}} + (-1)^{\lfloor{3\pi\rfloor}} + (-1)^{\lfloor{4\pi\rfloor}} = (-1) + 1 + (-1) + 1 = 0##
For:
##n = 1## and ##\alpha =e##, ##S_{1}=(e)=1##
##n = 2## and ##\alpha =e##, ##S_{2}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} = 1 + (-1) = 0##
##n = 3## and ##\alpha =e##, ##S_{3}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} + (-1)^{\lfloor{3e\rfloor}} = 1 + (-1) + 1 = 1 ##
##n = 4## and ##\alpha =e##, ##S_{4}=(e)= (-1)^{\lfloor{e\rfloor}} +(-1)^{\lfloor{2e\rfloor}} + (-1)^{\lfloor{3e\rfloor}} + (-1)^{\lfloor{4e\rfloor}} = 1 + (-1) + 1 + 1 = 2##
Thank you in advance.