Is a person allowed to sleep during the day of Rosh HaShanah? I have heard that you're not.
The Rama (Orach Chayim 583:2) says that some have the practice not to sleep on Rosh HaShanah, and it is a correct practice. The Taz (3) quotes the source of this: The Yerushalmi which says "Whoever sleeps on Rosh HaShanah, his luck sleeps." The Magen Avraham (6) quotes the Arizal, that it is permissible to sleep after midday, since the angel (of "luck," I guess) has already been awoken by then. However, the Taamei HaMinhagim #606 (in Kuntres Acharon #22) writes in the name of Mateh Efrayim that you shouldn't sleep even according to the Arizal, because ordinary people don't do even one thousandth of what the Arizal did even in his sleep.
The Magen Avraham there also adds that someone who sits idle is just as bad as if he were sleeping. The Machatzis HaShekel explains that the reason for the Yerushalmi is that the person isn't praying or doing something to awaken the angel for him. Therefore, being idle is not awakening the angel.