Hiya Sabina, (From British English, not American) I’ll try answer this as best I can. These phrases are horribly similar, and often used interchangeably, but do have differences in a lot of contexts. The best way to understand is to hear the phrases in real life, but I know that can’t always be that straightforward. I'm sorry, this is a long one, I hope it makes some sense. "Another day" at its simplest, is a contraction of "an other day". It basically means a different day: "Shall we meet up today?" "No, I can’t." "What's another (a different) day you can do?" "Tuesday?" "Sure, Tuesday is good." OR "Shall we go to the shop on Wednesday?" "I'm busy, can we go another (a different) day?" "Yeah, what about tomorrow?" "Yeah, I can do that." For "any other day", if it's used with the singular "day", it is generally an emphatic and/or informal way of saying "another day". Emphatic: "Shall we meet up today?" "No, *any* other day than today." (said with emphasis) "Why not?" "I really, really have to finish my homework assignment. I cannot do tomorrow." OR interchangeably (and informally) with “another day”: “Shall we meet up today?" “No, I can’t.” “Is there any other day…?” Here it is taken to mean “can you meet up another day/a different day”, but the wording used is closer to a shortened way of saying “are there any other days you can meet up” (see below). This is just the typical lazy/relaxedness of the way the language is used. You probably wouldn’t use the singular “day” in “is there any other day” in a formal situation, when you’re being careful with your grammar (it’s not the most correct grammar usage). You would say “are there any other days” for plural, or “is there another day” for singular. Traditionally/formally, it is used as "any other days", with the plural "days". "Shall we meet up today?" "No, I can’t." "Are there any other days you can meet up?" (“Are there any different days you can meet up?”) (CONT)

Another day - a different day or an additional day. Any other day - any day but the one we're talking about. Some other day - a different day.

Hello, "another day - a different and separate day usually in the future. "any other day - any different day in the past or future excluding a particular day. "some other day" - next time, not now. I hope this answers your question. If you’d like to practice speaking. I’d be happy to help you develop a study plan. Feel free to view my teaching profile and class offerings and send me a message if you think I can help you. Regards