“Hereafter, we let n and m refer to the number of non-compactified space and time dimensions, or more generally to the effective spacetime dimensionality that is relevant to the low-energy physics we will be discussing later.”
For example, in 2 dimensions embedded in our 3 the force potential decreasing with square distance would have given a hint to the 2-dim occupants that they are embedded in 3. We don't see the decreasing with cube, so we don't seems to be embedded in full-flat 4+. (even more - the force falls off linear with distance in galactics, i.e. "dark force"/MOND, which seems to be another indication of some holograghy-like "folding" 3->2 (the other being is the black hole holography))
https://en.wikipedia.org/wiki/Extra_dimensions