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A celebrated and deep result of Green and Tao states that the primes containarbitrarily long arithmetic progressions. In this note I provide astraightforward argument demonstrating that the primes get arbitrarily close toarbitrarily long arithmetic progressions. The argument also applies to `largesets' in the sense of Erd\H{o}s-Tur\'an. The proof is short, completelyself-contained, and aims to give a heuristic explanation of why the primes, andother large sets, possess arithmetic structure.

Almost arithmetic progressions in the primes and other large sets