The approximation holds only for b << 10^k. Then you don't get to take O(10^k) steps to work you way around the unit interval, do you? After the approximation breaks down, you don't know that the spacing is approximately uniform anymore.
The spacing is not uniform, it's just bounded by O(1/10^(k/2)). The spacing is, at its greatest, about 1/sqrt(2 log(10) 10^k).
If you knew the number of steps, I think you could only make a lower bound on spacing, not an upper bound. Since at worst all the steps could be clustered really close, except for the last one.
But now I have to think more because it seems like it should actually help the argument that the spacing gets finer and finer as you step, because that is what we want anyway. And is the number of steps actually important to the proof?
The number of steps tells you how big the last step is.