sort n numbers in the range [0,1] without multiplying or dividing
The $\Omega(n \log n)$ lower bound still applies; you still can't do better than $O(n \log n)$.

(Proof: Take any array of numbers.  You can find the maximum and minimum in $O(n)$ time.  Then, rescale the array.  In particular, if the minimum and maximum are $\alpha$ and $\beta$, replace each element $x$ of the array with $(x-\alpha)/(\beta-\alpha)$.  After the replacement, every element will be in the range $[0,1]$.  If you had a way to sort such arrays faster than $O(n \log n)$, you'd immediately obtain a way to sort the original array faster than $O(n \log n)$.)