On the dimension of iterated sumsets

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.