A fellow editor was wondering aloud (OK, on email) about a sentence with a construction similar to the above (What would result in you sounding better?). She thought perhaps it should be your rather than you: What would result in your sounding better? So… what would?
Both are in fact acceptable, although one is safer than the other. Some people are inclined to think that only the your version is correct, because they see becoming as a noun form (a gerund, to be precise) and thus determine that it should be modified by an adjectival form, which a possessive is. In this case, the complement of in is the noun phrase headed by becoming and with the modifier your and the complement one…
But we can see that the you version is quite commonly used, and is not generally questioned except by a few people, generally those who make it their business to question such things. So it must work somehow.
My unauthorized, off-the-top-of-my-head analysis of this is that in the you case, becoming is in fact a present participle, and (to be simple about it) you is its subject (note that because a participle is not a finite verb form, its subject is not in the nominative). This is similar to It was silly for him to do that, where him is the subject of the infinitive to do.
But, to go into the more complicated detail, the verb phrase headed by to do is actually the complement of him (and the verb phrase headed by sounding is actually the complement of you), and the real subject of the non-finite verb phrase is a silent repetition of the same pronoun. The explicit pronoun (you/him) is the complement of the preposition before it (in/of). The reason we talk of the silent repetition is that the verb phrase must have a subject but it is also the complement of the pronoun, and the complement of the preposition must be a noun phrase – and the reason for those “must”s is rather longer than I think you want to read here. The second instance of the pronoun is silent because it’s already been said once.
I’m not sure exactly how clear that is, because syntactic theory can be a bit recondite, but the short of it is that you’re safe with your in any case (and the syntactic analysis is easier to understand, so it’s not surprising that it’s preferred by sticklers), but you actually have a suitable case for using you if you prefer.
An illustration of participle versus gerund can be given by the following three sentences:
They were all alarmed by the sight of John curled up on the floor.
They were all alarmed by the sight of John curling up on the floor.
They were all alarmed by the sight of John’s curling up on the floor.
The first two are participles (past and present); the third is a gerund.
It has been pointed out that Henry Fowler decried this usage; his position was opposed by Otto Jespersen. Here are the first couple of examples from The New Fowler’s:
Women having the vote reduces men’s political power.
Women having the vote share political power.
The complaint against the first takes having as the subject of the sentence, and thus women as modifying it. Actually, however, it is the entire non-finite verbal phrase women having the vote that is the subject.
The second takes women as the subject, and having the vote as a participial phrase modifying women just as in They were all alarmed by the sight of John curling up on the floor. This is not the same case and can’t really be picked at one way or another (well, not for that reason – it’s an awkward sentence and could be rephrased as Women who have the vote…, using a relative to introduce an inflected verb).
Other examples are included as being adduced by both sides, and one catches my eye as adding to the matter: I set my heart on you coming to Spain. In that case, it is another instance of the whole participial phrase being treated as a unit, and I must say shows an example where the preposition does not require a direct noun object, since it is the action of coming rather than the you per se that the heart is set on. This can also parallel They were all alarmed by the sight of John curling up on the floor if it is the whole action rather than John himself that is shocking. (We may note an ambiguity: I set my heart on you coming to Spain could also be read to mean “Coming to Spain, I set my heart on you,” and in that case the syntactic structure is of course different – the participial phrase modifies the main verb.)
Fowler’s opponent in the matter – that is, the one who said such constructions were acceptable – was, as it happens, Otto Jespersen. Otto Jespersen was one of the great early leading lights in linguistics, widely respected by those who study the structure and function of language. Fowler is widely respected by the “I know my facts and don’t give me your fancy scientific analysis” set. (These same people dislike The New Fowler’s for its open-mindedness.)
In any event, The New Fowler’s presents many examples of the non-possessive construction, enough to demonstrate that those who declare it wrong are arguing in the face of the evidence and setting themselves above some of the language’s leading lights.
It is right to recognize and work with the fact that some people are uncomfortable with the non-possessive structure. It is a mistake to declare it ungrammatical. One might as well say that birds aren’t birds if we can’t identify what kind of birds they are.