On the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sums and products of subsets of Euclidean space

Kyle Hambrook, Krystal Taylor

Published 2018-10-26Version 1

We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form $RY +Z, $ where $R \subseteq (0,\infty)$ and $Y, Z \subset \mathbb{R}^d$. Most notable, for each $\alpha \in [0,1]$ and for each non-empty set $Y \subseteq \mathbb{R}$, we prove the existence of a compact set $R \subseteq (0,\infty)$ such that $\dim_H(R) = \dim_F(R) = \alpha$ and $\dim_F(RY) \geq \min\{ 1, \dim_F(R) + \dim_F(Y)\}$. This work is a contribution to the set of problems which study the measure and dimension of images of subsets of Euclidean space under Lipschitz maps.