Interior of sums of planar sets and curves

TL;DR

This paper investigates the interior of sums of planar sets and curves, showing conditions under which the sum has non-empty interior, and providing examples and methods related to sets with large measure, curvature, and Cantor sets.

Contribution

It introduces new conditions for the sum of planar sets and curves to have non-empty interior, including generalized product structures and curvature assumptions, and analyzes specific Cantor sets.

Findings

A large measure set can have an empty interior when summed with a circle.

Sum of certain product sets with curves of non-zero curvature has non-empty interior.

The pinned distance set of specific Cantor sets has non-empty interior.

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.