Jörg Schmeling: On the dimension of iterated sumsets

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_1+ ... + a_k : a_i ∈ A}.

We show that for any non-decreasing sequence { α_k }_{k=1}^∞ taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension α_k 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.