On the Dimension of Iterated Sumsets

Summary, in English

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: Jörg Schmeling; Pablo Shmerkin
• Department/s: Mathematics (Faculty of Engineering); Dynamical systems
• Publishing year: 2010
• Language: English
• Pages: 55-72
• Publication/Series: Recent Developments in Fractals and Related Fields
• Links: http://dx.doi.org/10.1007/978-0-8176-4888-6_5
• Document type: Conference paper
• Publisher: Birkhäuser Verlag
• Topic: Mathematics
• Conference name: Conference on Fractals and Related Fields
• Conference date: 0001-01-02
• Conference place: Monastir, Tunisia
• Status: Published
• Research group: Dynamical systems
• ISBN/ISSN/Other: ISBN: 978-0-8176-4887-9