On the Dimension of Iterated Sumsets

Jörg Schmeling, Pablo Shmerkin

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Sammanfattning

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.
Originalspråkengelska
Titel på värdpublikationRecent Developments in Fractals and Related Fields
FörlagBirkhäuser Verlag
Sidor55-72
ISBN (tryckt)978-0-8176-4887-9
DOI
StatusPublished - 2010
EvenemangConference on Fractals and Related Fields - Monastir, Tunisien
Varaktighet: 0001 jan. 2 → …

Konferens

KonferensConference on Fractals and Related Fields
Land/TerritoriumTunisien
OrtMonastir
Period0001/01/02 → …

Ämnesklassifikation (UKÄ)

  • Matematik

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