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 $\alpha \in [0,1]$ and for each non-empty compact set $R\subseteq (0,\infty )$, there exists a compact set $Y\subseteq [1,2]$ such that ${\dim }_F(Y)={\dim }_H(Y)=\overline{{\dim }_M}(Y)=\alpha$ and ${\dim }_F(RY)\ge \min \{1,{\dim }_F(R)+{\dim }_F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.