how to cover the unit interval by using shift operators

Question

Consider the following partition of $[0,1)$: $$P:=\left\{ [0,\frac{1}{2^n}), [\frac{1}{2^n},\frac{2}{2^n}),[\frac{2}{2^n},\frac{3}{2^n}), [\frac3{2^n},\frac{4}{2^n}), \cdots, [\frac{2^n-1}{2^n},1)\right\}.$$ Consider shift-operators $T_k:t\mapsto (t+k)\mod 1$. Let $I:=\cup_i A_i$, $A_i \in P$, with $m(I)=\frac{1}{2^j}$, where $m$ stands for the Lebesgue measure and $j\le n$. I want to choose some $T_k$'s such that $\cup_k T_k(I) =[0,1)$. How many $T_k$'s do I need to take? Of course, it is less than $2^n+1$ but I am wondering whether I can find a better estimate.