Interior of sums of planar sets and curves

Abstract: Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\left(A+\Gamma\right)^{\circ}$, when $\Gamma$ is a piecewise $\mathcal{C}^2$ curve and $A\subset \mathbb{R}^2.$ To begin, we give an example of a very large (full-measure, dense, $G_\delta$) set $A$ such that $\left(A+S^1\right)^{\circ}=\emptyset$, where $S^1$ denotes the unit circle. This suggests that merely the size of $A$ does not guarantee that $(A+S^1)^{\circ }\ne\emptyset$. If, however, we assume that $A$ is a kind of generalized product of two reasonably large sets, then $\left(A+\Gamma\right)^{\circ}\ne\emptyset$ whenever $\Gamma$ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of $C:=C_{\gamma}\times C_{\gamma}$, $\gamma \geq \frac{1}{3}$, pinned at any point of $C$ has non-empty interior, where $C_{\gamma}$ (see (1.1)) is the middle $1-2\gamma$ Cantor set (including the usual middle-third Cantor set, $C_{1/3}$). Our proof for the middle-third Cantor set requires a separate method. We also prove that $C+S^1$ has non-empty interior.