Hello!

Let {x} be the distance of x to the nearest integer value and let $\displaystyle x=0.\overline{b_{1}b_{2} \cdots b_{m}}$ be a binary expansion with repetition.

I want to prove that for all integers k and $\displaystyle k_{0}$ the following is valid:

$\displaystyle \{2^{mk + k_{0} } x\} = \{2^{k_{0} } x \}$

Well, in my opinion it is only valid for positive integers k and $\displaystyle k_{0}$.

Anyway, here is what I did:

$\displaystyle \{2^{mk + k_{0} } x\} = \{2^{mk + k_{0} } \cdot 0.\overline{b_{1}b_{2} \cdots b_{m}}\} = \{ 2^{k_{0}} \cdot 2^{mk} \cdot } 0.\overline{b_{1}b_{2} \cdots b_{m}} \} = \{ 2^{k_{0}} \cdot \cdot \sum_{i=1}^{m} \overline{b_{i}} 2^{-i + mk} \} $

Now some of these values in the sum must be integer values.

Or do I just have to say that there is going to be a k-times shift of m digits and because of the repetition of x it will be always the same value. I just donīt know how to represent that integer value in the sum which will then disappear because of the definition of {}.

Does anyone have any advice? Thanks!

Regards