Self similar sets, entropy and additive combinatorics

This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in the original proof but also falls short of a complete proof. In the last section we discuss how the heuristic argument is turned into a rigorous one.

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