TY - JOUR

T1 - On self-similar sets with overlaps and inverse theorems for entropy

AU - Hochman, Michael

PY - 2014

Y1 - 2014

N2 - We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of minf1; sg, where s is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coeffcients. As applications we prove Furstenberg's conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2-n has only a small amount of growth under convolution.

AB - We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of minf1; sg, where s is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and confirms it when the generating similarities have algebraic coeffcients. As applications we prove Furstenberg's conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2-n has only a small amount of growth under convolution.

UR - http://www.scopus.com/inward/record.url?scp=84899667915&partnerID=8YFLogxK

U2 - 10.4007/annals.2014.180.2.7

DO - 10.4007/annals.2014.180.2.7

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AN - SCOPUS:84899667915

SN - 0003-486X

VL - 180

SP - 773

EP - 822

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -