On the dimension of iterated sumsets

Schmeling, Jörg and Shmerkin, Pablo (2009) On the dimension of iterated sumsets. [MIMS Preprint]

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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+...+A (k times). We show that for any non-decreasing sequence {a_k} taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension a_k for all k. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets. These results are in stark contrast to the Plunnecke-Rusza inequalities in additive combinatorics. However, for lower box-counting dimension, the analogue of the Plunnecke-Rusza inequalities does hold.

Item Type: MIMS Preprint
Uncontrolled Keywords: CICADA, Hausdorff dimension, box dimension, sumsets, Plünnecke-Rusza inequality
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 05 Combinatorics
MSC 2010, the AMS's Mathematics Subject Classification > 28 Measure and integration
Depositing User: Mr Pablo Shmerkin
Date Deposited: 14 Oct 2009
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1332

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