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On the Dimension of Iterated Sumsets

Schmeling, Jörg LU and Shmerkin, Pablo (2010) Conference on Fractals and Related Fields In Recent Developments in Fractals and Related Fields p.55-72
Abstract
Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = {a(1) ... + a(k) : a(i) is an element of A}. We show that for any nondecreasing sequence {alpha(k)}(k=1)(infinity) taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension ak for all k >= 1. We also show how to control various kinds of dimensions simultaneously for families of iterated sumsets. These results are in stark contrast to the Plunnecke-Ruzsa inequalities in additive combinatorics. However, for lower box-counting dimensions, the analog of the Pliinnecke Ruzsa inequalities does hold.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Recent Developments in Fractals and Related Fields
pages
55 - 72
publisher
Birkhaüser
conference name
Conference on Fractals and Related Fields
external identifiers
  • wos:000289340300005
ISBN
978-0-8176-4887-9
DOI
10.1007/978-0-8176-4888-6_5
language
English
LU publication?
yes
id
17dd09ab-bb62-429b-9b84-97c6c3d61f82 (old id 1964024)
date added to LUP
2011-05-19 15:58:43
date last changed
2017-03-16 14:24:03
@inproceedings{17dd09ab-bb62-429b-9b84-97c6c3d61f82,
  abstract     = {Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = {a(1) ... + a(k) : a(i) is an element of A}. We show that for any nondecreasing sequence {alpha(k)}(k=1)(infinity) taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension ak for all k >= 1. We also show how to control various kinds of dimensions simultaneously for families of iterated sumsets. These results are in stark contrast to the Plunnecke-Ruzsa inequalities in additive combinatorics. However, for lower box-counting dimensions, the analog of the Pliinnecke Ruzsa inequalities does hold.},
  author       = {Schmeling, Jörg and Shmerkin, Pablo},
  booktitle    = {Recent Developments in Fractals and Related Fields},
  isbn         = {978-0-8176-4887-9},
  language     = {eng},
  pages        = {55--72},
  publisher    = {Birkhaüser},
  title        = {On the Dimension of Iterated Sumsets},
  url          = {http://dx.doi.org/10.1007/978-0-8176-4888-6_5},
  year         = {2010},
}