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