Self Similar Sets, Entropy and Additive Combinatorics

Michael Hochman

doi:10.1007/978-3-662-43920-3_8

Self Similar Sets, Entropy and Additive Combinatorics

Michael Hochman^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

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.

Original language	English
Title of host publication	Geometry and Analysis of Fractals
Publisher	Springer New York LLC
Pages	225-252
Number of pages	28
ISBN (Print)	9783662439197
DOIs	https://doi.org/10.1007/978-3-662-43920-3_8
State	Published - 2014
Event	International Conference on Advances of Fractals and Related Topics - , Hong Kong Duration: 10 Dec 2012 → 14 Dec 2014

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	88
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Advances of Fractals and Related Topics
Country/Territory	Hong Kong
Period	10/12/12 → 14/12/14

Bibliographical note

Funding Information:
Supported by ERC grant 306494

Access to Document

10.1007/978-3-662-43920-3_8

Cite this

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title = "Self Similar Sets, Entropy and Additive Combinatorics",

abstract = "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.",

author = "Michael Hochman",

note = "Funding Information: Supported by ERC grant 306494 ; International Conference on Advances of Fractals and Related Topics ; Conference date: 10-12-2012 Through 14-12-2014",

year = "2014",

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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.

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T2 - International Conference on Advances of Fractals and Related Topics

Y2 - 10 December 2012 through 14 December 2014

ER -

Self Similar Sets, Entropy and Additive Combinatorics

Abstract

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