In what case to use the last (with the definite article) with time periods that precede the current time period?

Use last without the or a preposition when talking about a time period that precedes the current time period. However, when talking about several time periods that precede the current time period (in this example: several days), use the last and the respective preposition: e.g. over the last.

For example, a phrase like this: His condition has severely degraded over the last few days.
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