Measure and dimension of sums and products

Abstract:

We investigate the Fourier dimension, Hausdorff dimension, and Lebesgue measure of sets of the form $RY + Z,$ where $R$ is a set of scalars and $Y, Z$ are subsets of Euclidean space. Regarding the Fourier dimension, we prove that for each $\alpha \in [0,1]$ and for each non-empty compact set of scalars $R \subseteq (0,\infty )$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim _F(Y) = \dim _H(Y) = \overline {\dim _M}(Y) = \alpha$ and $\dim _F(RY) \geq \min \{ 1, \dim _F(R) + \dim _F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones. Further, we investigate lower bounds on the measure and dimension of $RY+Z$ for $R\subset (0,\infty )$ and arbitrary $Y$ and $Z$; these latter results provide a generalized variant of some theorems of Wolff and Oberlin in which $Y$ is the unit sphere.