Self similar sets, entropy and additive combinatorics
arXiv:1307.6399 · doi:10.1007/978-3-662-43920-3_8
Abstract
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.
21 pages, 2 figures; submitted to Proceedings of AFRT 2012. v5: more typos corrected