On the Dimension of Iterated Sumsets
(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:
https://lup.lub.lu.se/record/1964024
- author
- Schmeling, Jörg LU and Shmerkin, Pablo
- organization
- publishing date
- 2010
- 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
- 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
- 2024-01-12 18:48:16
@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}}, 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}}, }