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In this lecture, we describe the simple but fundamental *Furstenberg correspondence principle* which connects the “soft analysis” subject of ergodic theory (in particular, recurrence theorems) with the “hard analysis” subject of combinatorial number theory (or more generally with results of “density Ramsey theory” type). Rather than try to set up the most general and abstract version of this principle, we shall instead study the canonical example of this principle in action, namely the equating of the *Furstenberg multiple recurrence theorem* with Szemerédi’s theorem on arithmetic progressions.

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