1. ## About accumulation points of a set in [0,1]

Hi!

If $\displaystyle \lambda>0$:
What are the accumulation points of $\displaystyle A=\left\{\lambda n-[\lambda n]:n\in\mathbb{N}\right\}$ in $\displaystyle [0,1]$?

2. ## Re: About accumulation points of a set in [0,1]

Is $\displaystyle [\lambda n]$ the floor function?

3. ## Re: About accumulation points of a set in [0,1]

Originally Posted by Gusbob
Is $\displaystyle [\lambda n]$ the floor function?
Yes.

4. ## Re: About accumulation points of a set in [0,1]

Originally Posted by Gusbob
Is $\displaystyle [\lambda n]$ the floor function?
It far more than that. Look at the webpage.

Originally Posted by eraldcoil
If $\displaystyle \lambda>0$:
What are the accumulation points of $\displaystyle A=\left\{\lambda n-[\lambda n]:n\in\mathbb{N}\right\}$ in $\displaystyle [0,1]$?
Well at least I have no worry that Soroban will give a complete solution to this one.

This is a great teaching question. DO IT!

First let $\displaystyle \lambda =\tfrac{1}{2}$ then $\displaystyle \lambda =\tfrac{1}{4}$ then $\displaystyle \lambda =\tfrac{3}{5}$.

Now what happens if $\displaystyle \lambda =\tfrac{\pi}{2}~?$

If you go back to that webpage, look at the graphs, what is going on?

### accumilation points of the set (0 ,1]

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