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

• Apr 11th 2013, 03:32 PM
eraldcoil
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 $\displaystyle [0,1]$?

• Apr 11th 2013, 04:28 PM
Gusbob
Re: About accumulation points of a set in [0,1]
Is $\displaystyle [\lambda n]$ the floor function?
• Apr 11th 2013, 04:42 PM
eraldcoil
Re: About accumulation points of a set in [0,1]
Quote:

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

Yes.
• Apr 11th 2013, 05:01 PM
Plato
Re: About accumulation points of a set in [0,1]
Quote:

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

It far more than that. Look at the webpage.

Quote:

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?