On the Dimension of Iterated Sumsets

  • Jörg Schmeling
  • Pablo Shmerkin
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as
$$kA =\{ {a}_{1} + \cdots + {a}_{k} : {a}_{i} \in A\}.$$
We show that for any nondecreasing sequence {α k }k = 1 taking values in [0, 1], there exists a compact set A such that kA has Hausdorff dimension α k 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 Plünnecke–Ruzsa inequalities in additive combinatorics. However, for lower box-counting dimensions, the analog of the Plünnecke–Ruzsa inequalities does hold.


Fractal Dimension Additive Combinatorics Binary Digit Nondecreasing Sequence Dyadic Interval 
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Both authors were members of the Mathematical Sciences Research Institute when this project was started. We are grateful to MSRI for its fruitful atmosphere and financial support. P.S. acknowledges support from EPSRC grant EP/E050441/1 and the University of Manchester.


  1. 1.
    Kenneth Falconer. Fractal geometry: Mathematical foundations and applications. Wiley, Chichester, 1990.Google Scholar
  2. 2.
    Terence Tao and Van Vu. Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund University, Lunds Institute of TechnologyLundSweden
  2. 2.Centre for Interdisciplinary Computational and Dynamical Analysis and School of Mathematics, Alan Turing BuildingUniversity of ManchesterManchesterUK

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