TY - GEN
T1 - Self Similar Sets, Entropy and Additive Combinatorics
AU - Hochman, Michael
N1 - Funding Information: Supported by ERC grant 306494
PY - 2014
Y1 - 2014
N2 - This article is an exposition of the main result of [Hoc12], that self-similar sets whose dimension is smaller than the trivial upper bound have "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 [Hoc12], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.
AB - This article is an exposition of the main result of [Hoc12], that self-similar sets whose dimension is smaller than the trivial upper bound have "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 [Hoc12], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.
UR - http://www.scopus.com/inward/record.url?scp=84906857909&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/978-3-662-43920-3_8
DO - https://doi.org/10.1007/978-3-662-43920-3_8
M3 - منشور من مؤتمر
SN - 9783662439197
T3 - Springer Proceedings in Mathematics and Statistics
SP - 225
EP - 252
BT - Geometry and Analysis of Fractals
T2 - International Conference on Advances of Fractals and Related Topics
Y2 - 10 December 2012 through 14 December 2014
ER -