"of their being" vs "of there being"

I am reading a BBC article on "Why pressing ‘upload’ means losing your rights". On page 2 of that article there is a paragraph that reads:

Meanwhile, legal thinking on digital rights is slowly catching up with the absurdity of their being almost no current recourse for loss, deletion or the whims of a service provider; but its pace is massively exceeded by the rate at which material is flowing online, into the hands of businesses whose profitability rests on owning and exploiting everything you give them.

My question is what is the difference between "of their being" and "of there being", and are both acceptable and when? Both have millions of results on Google. I know their is used for possession but I don't understand how that fits in here.

Thank you

Solution 1:

It should definitely be "of there being" in the referenced paragraph.

Let us illustrate by providing possible alternative wordings:

1. A consequence of their being is ...
2. A consequence of their being available is ...
3. A consequence of there being a solution is ...

These can be rephrased as:

1. A consequence of their existence is ...
2. A consequence of their availability is ...
3. A consequence of the existence of a solution is ...

So what is going on?

• "Their being", and "their being [adjective]" refer to the existence, respectively the property expressed by the adjective, of the subject referenced by "their".

• "There being", on the other hand, indicates a general state of affairs.

I hope that clarifies the matter for you.

Solution 2:

Good question, I agree that it's an error.

Possessive adjectives are used before gerunds (My coming surprised everyone) and not object pronouns (Me coming surprised everyone).

This latter is heard more and more and thus represents a change in progress.

But there is no possessive adjective for there and in the sentence, there is no logical reference for the their, so it seems to be an oversight in the editing.