Which of the phrases is correct and why? "This document assumes a general familiarity with ..." or "This document assumes general familiarity with ..." Google search gives me approximately equal distribution of occurrences, so I don't know which one is better. Would greatly appreciate your assistance.
In OP's specific case, I think omitting the article is fine (as she says herself, both versions occur about equally).
But in related variants like "I assume you have a firm grasp of English grammar", you actually need the article, to sound like a native. So if you're not sure in any given case, include a/an.
David Crystal, probably Britain's most renowned linguist, has written a helpful article on this thorny issue.
Crystal refers to Quirk's (A Comprehensive Grammar of the English Language) analysis of when the indefinite article can be used in front of a modified uncountable noun. Crystal cites Quirk as follows:
The conditions under which a/an occurs in such cases are unclear.
However, there are two useful patterns that Quirk has identified. Firstly, it seems that the article is used if the noun "refers to a quality or other abstraction which is attributed to a person." So,
This document assumes a general familiarity with ...
is fine because familiarity is a quality attributed to a person. But the indefinite article is not correct in the following:
*We made an important progress ...
since progress is not such a personal attribute.
Secondly, Crystal summarises Quirk as follows:
... the greater the amount of premodification or postmodification, the more likely we will find the indefinite article.
She played the oboe with sensitivity.
?She played the oboe with a sensitivity.
She played the oboe with a great sensitivity.
She played the oboe with a great and engaging sensitivity.
She played the oboe with a sensitivity that delighted the critics.
She played the oboe with a great and engaging sensitivity that delighted the critics.
Crystal then goes on:
Having said all that, I'm not entirely sure which uncountables follow these trends. The semantic criterion (personal attribution) is inevitably a bit fuzzy. Is plagiarism included, for example? Would you accept The teacher discovered a fresh plagiarism? I think there might be quite a lot of divided usage here.
The basic advice I get from this for English language learners is they should be aware of Quirk's two "rules". And in fuzzy cases such as plagiarism neither use nor non-use of the indefinite article is likely to have native speakers throwing up their hand in horror at an egregious grammar mistake.