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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.9 Ergodic Theorems

Vitaly Bergelson

Consider the sequence image, where z is a complex number of modulus 1. While for z ≠ 1 our sequence is not convergent, it is not hard to see that, on average, it exhibits quite regular behavior. Indeed, using the formula for the sum of a geometric progression, and assuming that z ≠ 1, we have, for any N > M ≥ 0,

Image

which implies that when N - M is large enough, the averages

Image

are small. More formally, we have

This simple ...

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