On the Dimension of Iterated Sumsets
نویسندگان
چکیده
kA = {a1 + . . .+ ak : ai ∈ A}. We show that for any non-decreasing sequence {αk} ∞ k=1 taking values in [0, 1], there exists a compact set A such that kA has Hausdorff dimension αk for all k ≥ 1. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets. These results are in stark contrast to the Plünnecke-Rusza inequalities in additive combinatorics. However, for lower box-counting dimension, the analogue of the Plünnecke-Rusza inequalities does hold.
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تاریخ انتشار 2009