ちかい vs ちかくの - what's the difference?
"I put the letter into a mailbox near my house" is translated as
てがみを いえのちかくの ポストに 入れました。
I know that in other sentences Duolingo has used ちかい instead of ちかくの for "nearby".
Is there a difference in meaning / nuance, or can I use either?
ちかい is an i-adj. ちかく is an adverb and an adverbial noun so you can also use it in front of a noun by adding the particle の. Both ちかいポスト and ちかくの ポスト are correct. It's just two different ways of saying it.
Thank you. (Btw, what would an adverbial noun be? Can this be expressed in English?)
In English, an adverbial noun is a noun that is used as an adverb. For example, "I am going home" or "I'll see you tomorrow." In these two sentences, home and tomorrow are adverbial nouns.
In Japanese, adverbs can be used as nouns. Perhaps it would be better to call them nominal adverbs rather than adverbial nouns, but the point is that the word ちかく can mean "a nearby place," so ちかくのポスト is a nearby mailbox. The の is necessary here because ちかく is being used as a noun.
There are a lot of adjectives can be used this way. おおくの many おおくのひと many people とおくの distant とおくのしま a distant island