It was recently proved that the prime numbers contain arbitrarily long arithmetic progressions. We describe this result from a "multiscale analysis" perspective. The first "scale" is the division of integers into even and odd integers, with almost all the primes of course falling into the latter category. The next scale is then formed by subdividing further based on the residue classes modulo three, and so forth. After a while, we obtain the "almost primes" - those numbers with no small prime factors, which are quite well understood. But to pass from understanding the set of almost primes to understanding the smaller set of primes is still a very difficult problem. To obtain arithmetic progressions, one can avoid this problem by using ideas from ergodic theory to create further (non-trivial) scales to analyze the primes, which "factor out" all the structure from the primes and replace the (indicator function of the) primes by a smoother function which is an ergodic "average" of the primes, plus an error term which is "random" in a certain sense and thus negligible. This ergodic "homogenization" procedure may well have applications to other multiscale analysis problems.
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