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2009.72: On the dimension of iterated sumsets

2009.72: Jörg Schmeling and Pablo Shmerkin (2009) On the dimension of iterated sumsets.

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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 2000 > 05 Combinatorics
MSC 2000 > 28 Measure and integration
MIMS number:2009.72
Deposited By:Mr Pablo Shmerkin
Deposited On:14 October 2009

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