I don't get the logic of using - or not using - the article a in the examples below. Is there any rule on this score? Or is it grammatically allowed to both put or omit a in the cases like these?
After a few minutes of (a) futile argument we had to accept...
He has a reputation of (a) troublemaker.
When you're unsure which article to use, or whether to use an article at all, it's good to start with looking at the noun: is it countable, or is it uncountable? Is it singular or plural?
After a few minutes of (a) futile argument we had to accept...
The noun "argument" is a noncount noun here, and means "the process of discussing something, arguing over something".
Hence, it's better not to use "a" there, because the indefinite article is used with singular count nouns.
Compare:
After several hours of fighting the enemy withdrew.
Your "argument" is close in meaning to "fighting".
The word "argument" has countable senses too: "a fact or statement used to support a proposition; a reason":
It is hard to imagine the court would seriously consider such an argument against otherwise-permissible regulation of guns.
He has the reputation of (a) troublemaker.
"Troublemaker" is a singular count noun, and we can use "a" here. We cannot just omit the use of articles here.
We cannot use "the", because we have in mind a "prototypical troublemaker".
Also note that you should use "the" before "reputation", because otherwise it might mean that the prototypical troublemaker has several reputations, and your subject has one of the several possible reputations.
The indefinite article instantiates an unidentified (anonymous) member of an abstract class.
A quantifier instantiates n members of an abstract class. The quantifier's complement specifies which class.
You can think of an operator like few as allocating memory for a small collection of instances of an abstract class, with the allocator's complement specifying which class that will be; the collection remains empty until its members are instantiated by an article.
After several quantifier minutesquantifier's complement we gave up.
After a instantiator few allocator minutes allocator's complement we gave up.
After a instantiator boatload allocator of problems allocator's complement we decided to roll back the nightly O/S update.
