On the dimension of iterated sumsets

Details On the dimension of iterated sumsets On the dimension of iterated sumsets

2009-06-08

arxiv.org/abs/0906.1537v2

Mathematics

Classical Analysis and ODEs

To appear in the Proceedings of the Conference on Fractals and Related Fields, Monastir, 2007

Scientific paper

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.

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