Doesn't the $p$ depend on $m$

Question

Consider the following text:

where $p \in \mathbb Z$. Can you tell me please: doesn't $p$ depend on the $m$? So it is preferable maybe to write $p_m$? Or does $p$ not depend on $m$?

Answer 1

Indeed $p$ does depend on $m$, but adding the extra subscript as you request would be unnecessary notation. The $p$ is not important, what's important is that $x$ is in an interval of width $\displaystyle\frac{1}{2^m}$.