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 (Formula presented) We show that for any nondecreasing 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 dimensions simultaneously for families of iterated sumsets. These results are in stark contrast to the Plünnecke–Ruzsa inequalities in additive combinatorics. However, for lower box-counting dimensions, the analog of the Plünnecke–Ruzsa inequalities does hold.