最近 and 先日: recent times and the other day
Distinguish a recent stretch of time from a particular unspecified day in the past.
Recent trends and recent events
最近 is read さいきん and refers to recent times up to the present. 「最近、よく歩いています」 describes walking frequently these days.
It can also refer to a single recent event, as in 「最近、新しい店ができました」, meaning that a new shop recently opened. 最近 is not limited to repeated actions.
Referring to a past occasion
先日 is read せんじつ and refers to a past day or occasion without specifying its date. 「先日はありがとうございました」 thanks someone for the other day.
「先日、駅で先生に会いました」 describes meeting the teacher at the station the other day. It is a completed event, not an announcement of a future meeting.
Different time scopes
「最近忙しいです」 broadly describes being busy these days. 「先日は忙しかったです」 narrows the focus to being busy on a particular recent occasion.
Neither word guarantees a fixed number of days. The scope of 'recent' depends on context, so check the date separately when precision matters.
Can 「先日」 be translated as 'the next day'?
Show the answer and why
No. It refers to a day or occasion already in the past. 「先日お送りしました」 says that something was sent the other day, not that it will be sent in the future.