I do not quite understand the proposed idea.
If space is created by just local presence of matter, then there should be little space where there is little matter, is this correct?
Also, if extra spatial dimensions are big, they are different than the spatial dimensions we already know, othervise we would call our space 4, 5, etc.- dimensional. Therefore, can we call those DIFFERENT dimensions (like 4th space dimension bent by gravity) indeed to be equal space dimensions? We have to assume then that some objects (gravitons) CAN exist in higher dimensions, but some other (protons) CAN NOT.
Small dimensions are different matter - if they are so small that we see them only from large distance, we may never "feel" them at all.
By the way, entangled particles do not need extra dimensions to be entangled. There may exist more simple explanation of their entanglement.
For example dig two holes - one on Earth, one on Mars, then take a coin and ask you friend to plant it EITHER in Earth OR in Mars hole, and without telling you in which one. Also ask him to disappear from anybody's sight, so noone can guess where the coin might be hidden.
Then you advertise and demonstrate (for a small donation to promote logic and science) a phenomenon of "FTL (faster than light)teleportation of information" or you may call it "FTL collapce of wave function" or even "Superluminal communication" (whatever you label it) as follows.
You claim, that you will INSTANTLY tell if there is a coin on Mars by simply digging the Earth hole.Unbelivable? May be. Also you claim that the coin "probability wave function" which is now 50% here and 50% on Mars will instantly (FTL) and magically collapse right before the public eyes. Public, of course, does not believe you (after all, nothing can travel or collapce faster than light, right?).
Then after short fanfare prelude (a drum man keeps beating), in a spotlight you dig the Earth hole, and upon finding a coin (or absence of it) -and viva -you IMMEDIATELY tell if there is a coin on Mars (!!!).
Public is shocked: how can one possibly know what is going on on Mars, and especially, that fast (way faster than light)? So they send someone to Mars to check, and indeed - you are right! (No money back).
Also, you point on the fact that right after you dug the Earth hole, the wave function of the coin collapsed FTL from 50/50 to 0/100 state! Again, the public does not believe and checks by numerous experiments. And again, you are right: while before unearthing propability to find coin was 50/50, right after that it became 0/100 (or 100/0) consistently test after test experiments.
This may sound unbelivable, but you can test this idea - it should work fine.
So, you do not need extra dimensions for entangled particles or "FTL teleportation of information".