On the dimension of iterated sumsets

TL;DR

This paper investigates the fractal dimensions of iterated sumsets of subsets of the real line, demonstrating the ability to prescribe dimensions of these sumsets independently, contrasting with classical additive combinatorics inequalities.

Contribution

It constructs sets with prescribed Hausdorff dimensions for all iterated sumsets, showing flexibility in fractal dimensions unlike traditional additive combinatorics results.

Findings

Existence of sets with prescribed Hausdorff dimensions for all iterated sumsets.

Control over multiple dimensions simultaneously for families of sumsets.

Lower box-counting dimension satisfies Plunnecke-Rusza inequalities.

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.