Dimension and measure of sums of planar sets and curves

Abstract

Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of $A+\Gamma:=\left\{a+v:a\in A, v\in \Gamma \right\}$ when $A\subset \mathbb{R}^2$ and $\Gamma$ is a piecewise $\mathcal{C}^2$ curve. Assuming $\Gamma$ has non-vanishing curvature, we verify that (a) if $\dim_{\rm H} A \leq 1$, then $\dim_{\rm H} (A+\Gamma)=\dim_{\rm H} A +1$, where $\dim_{\rm H}$ denotes the Hausdorff dimension; (b) if $\dim_{\rm H} A>1$, then $Leb_2(A+\Gamma)>0$, where $Leb_2$ denotes the $2$-dimensional Lebesgue measure; (c) if $\dim_{\rm H} A=1$ and $H^1(A) < \infty$, then $Leb_2(A+\Gamma)=0$ if and only if $A$ is an irregular (purely unrectifiable) $1$-set. Here, $H^1$ denotes the $1$-dimensional Hausdorff measure. Items (a) and (b) follow from previous works of Wolff and Oberlin using Fourier analysis. In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular $1$-set has $1$-dimensional Lebesgue measure equal to zero at almost every pin $t\in \mathbb{R}^2$.