 Re: the preposition befor DAY
Why should it be "in" for arrive? Arrive, unlike participate, is not a prepositional verb and doesn't have an associated preposition. It's an ordinary non-transitive one word verb. Because of the meaning it will often be followed by a prepositional phrase : "arrive" implies a place and a moment in time, and prepositions express location and time. But which preposition is used is dependent on the following noun, not on "arrive" :
We arrived / at the station
We arrived / on time
We arrived / in London
We arrived / behind schedule
We arrived / on the island
We arrived / at ten o'clock
Fall has arrived /above the treeline
... the boats arrived / below the fort
etc etc
Or of course there may be no prepositional phrase at all :
We arrived early
We arrived and checked in |