Measure and Dimension of Sums and Products

Abstract

We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form $RY + Z, $ where $R \subseteq (0,\infty)$ and $Y, Z \subseteq \mathbb{R}^d$. We prove a theorem on the Lebesgue measure and Hausdorff dimension of $RY+Z$; The theorem is a generalized variant of some theorems of Wolff and Oberlin in which $Y$ is the unit sphere, but its proof is much simpler. We also prove a deeper existence theorem: For each $α\in [0,1]$ and for each non-empty compact set $R \subseteq (0,\infty)$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim_F(Y) = \dim_H(Y) = \overline{\dim_M}(Y) = α$ and $\dim_F(RY) \geq \min\{ 1, \dim_F(R) + \dim_F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.