39 research outputs found
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.Comment: 21 pages, 2 figures; submitted to Proceedings of AFRT 2012. v5: more
typos correcte