Hello!

Let {x} be the distance of x to the nearest integer value and let be a binary expansion with repetition.

I want to prove that for all integers k and the following is valid:

Well, in my opinion it is only valid for positive integers k and .

Anyway, here is what I did:

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