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On the Dimension of Iterated Sumsets

Schmeling, Jörg LU and Shmerkin, Pablo (2010) Conference on 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
and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
Recent Developments in Fractals and Related Fields
pages
55 - 72
publisher
Birkhäuser Verlag
conference name
Conference on Fractals and Related Fields
conference location
Monastir, Tunisia
conference dates
0001-01-02
external identifiers
  • wos:000289340300005
  • scopus:85015330778
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
2016-04-04 10:21:16
date last changed
2022-03-15 21:38:40
@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    = {{Birkhäuser Verlag}},
  title        = {{On the Dimension of Iterated Sumsets}},
  url          = {{http://dx.doi.org/10.1007/978-0-8176-4888-6_5}},
  doi          = {{10.1007/978-0-8176-4888-6_5}},
  year         = {{2010}},
}