The ergodic theoretical proof of Szemerédi's theorem

Partial results were obtained previously by K. F. Roth (1952) who established the existence of arithmetic progressions of length three in subsets of Z of positive upper density, and by E. Szemeredi (1969) who proved the existence of progressions of length four. In 1976 Furstenberg noticed that the statement of Theorem I is equivalent to a statement about "multiple recurrence" of measure-preserving transformations, namely