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Events in the past are easy: all you really need is UTC time, but if you store local time and the offset from UTC (and optionally the tzdata timezone name) that works too.

For dates in the future, you need to store local time, the timezone name, and the calculated offset for that time. When the database changes, you need to check if the offset changes, and if so, think hard about what that means. Also, some future events are more appropriately scheduled for local time wherever the person happens to be, which is tricky too.