Measure and Dimension of Sums and Products

TL;DR

This paper studies the measure and dimension properties of sets formed by sums and products of sets in Euclidean space, providing new theorems on Lebesgue measure, Hausdorff, and Fourier dimensions, and constructing sets with prescribed dimensions.

Contribution

It introduces a simplified proof of existing theorems on measure and dimension of sum sets and proves a new existence theorem for sets with specific Fourier and Hausdorff dimensions.

Findings

Proved a theorem on Lebesgue measure and Hausdorff dimension of sum sets.

Established the existence of sets with prescribed Fourier and Hausdorff dimensions.

Demonstrated how to generate new Salem sets from existing ones.

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 $\dim_F(RY) \geq \min\{ 1, \dim_F(R) + \dim_F(Y)\}$. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.