Should "gerund + objective" be modified by adjectives or adverbs?

Form versus Function

This is a perennial confusion, one deriving in part from different sources using the word “gerund” in conflicting and contradictory ways, some of which are based in older analyses that no longer hold, others which are simply too fuzzy for practical application. You seem to be using “gerund” to mean any old ‑ing word at all, no matter what its part of speech is. That’s going to lead to confusion.

A word like studying, waiting, broadening, questioning, or giving is a regular non-finite verbal inflection, but that only tells you the word’s form not its function. It doesn’t tell you what it’s being used for because without the grammar of a surrounding phrase and how that word fits into that phrase grammatically, it in fact isn’t being used for anything at all and so has no part of speech.

Yet.

It may surprise you to learn that “gerund” isn’t so useful a term as you might think, and you don’t even need it. You’ll find that the analysis becomes far easier, both in this case and in more complex ones, if you discard the term entirely and stick strictly to parts of speech: verb, noun, adjective, adverb. If you want to discuss its broader syntactic role in the grammar as a constituent, then we use other terms for those constituents than parts of speech.

Instead, here is a simple guideline for classifying VERB-ing words into one of noun, verb, or adjective:

1. When the ‑ing word is a VERB, it can be modified by adverbs (that aren’t actually intensifiers instead like very), but not by adjectives. It can also take objects if it’s a transitive verb:

   • “Quickly giving her the day off was the best solution.”
   • “I was carefully giving her the delicate soufflé when her phone rang.”
   • “These stickers are for quickly giving your students’ papers an attractive decoration.”

2. When the ‑ing word is a NOUN, it can be modified by adjectives and quantifiers, but not by adverbs or intensifiers. It can often be inflected into the plural as well:

   • “Voluntary givings at churches during Christmastime are key to our global relief effort.”
   • “Any voluntary giving should be deducted from your taxes.”
   • “Here we call our donations bins our ‘giving boxes’, so please place your gifts in any of the three colorfully decorated giving boxes near the entrance.”

3. When the ‑ing word is an ADJECTIVE, it can be modified not only by adverbs (and not by other adjectives), but also by very and related words of its class (intensifiers):

   • “She was an endlessly giving person, even after the crooks took advantage of her.”
   • “She was a very giving person to her dying day, and beyond.”

The Latin term “gerund” isn’t a very good one for English for many reasons I won’t be repeating here. The important thing to remember is that “gerund” isn’t somehow its own part of speech: an ‑ing word derived from a verb is still always going to be one of either a verb or a noun or an adjective. Sometimes these latter two are referred to as verbal or deverbal nouns or adjectives, or as participial adjectives, to show that they’ve stopped being verbs.

The only ‑ing words that are doing a “gerundial” job are those which are still verbs, not deverbal nouns, and which happen to be acting as a substantive, meaning places in the grammar where a noun phrase is required — typically when a subject or object is called for.

SEE ALSO

• Is “running” a gerund or a participial adjective?
• Can a gerund be modified by an adjective?
• “I hate Jill singing those songs.” = “I hate Jill when she is singing those songs.”?
• What part of speech is ‘mountains’ in the sentence “I like climbing mountains”?
• How many parts of speech can a word be at the same time?
• When does a gerund become a verb?

To supplement the excellent answer from tchrist, I'll answer your question:

For example, is it OK to say "With my persistent broadening the horizon of my knowledge of cosmology, my interest in it is ever growing."?

No, it is not okay. "broadening" is apparently a noun here, since it is modified by the adjective "persistent" (as you recognize). But "the horizon of my knowledge of cosmology" is a NP (noun phrase) which is the direct object of "broadening". In English, nouns cannot have direct objects. This is a contradiction, because "broadening" can either be a noun or a verb, but it can't be both simultaneously.

Consequently, there are two ways of amending your example sentence: (1) change it so that "broadening" is unambiguously a noun, or (2) change it so that "broadening" is unambiguously a verb. In your discussion, you suggest (2), where the adjective "persistent" has been changed to an adverb, "persistently". Then "broadening" can be a verb modified by an adverb and taking a direct object.

Alternatively, (1), an "of" can be inserted to convert the NP "the horizon of my knowledge of cosmology" into the PP (prepositional phrase) "of the horizon of my knowledge of cosmology":

With my persistent broadening of the horizon of my knowledge of cosmology, my interest in it is ever growing.

This works, because although English nouns cannot have direct objects, they can take PP complements.

This sort of example cannot be correctly understood in traditional grammar, or in its modern offshoot dependency grammar, because it requires analysis in terms of multi-word phrases, not just a classification of words.

Footnote: By "traditional grammar" above, I mean a simple parts-of-speech analysis of the sort invented by the ancient Greeks. However, it would not cause any difficulty for the great traditional grammarians of the 20th century Otto Jespersen or Hendrick Poutsma.