On arithmetic sums of connected sets in $\mathbb{R}^2$

TL;DR

This paper proves that the sum of two connected sets in , where one is compact and not a line segment, always has a non-empty interior, extending previous results by relaxing compactness assumptions.

Contribution

It extends prior work by showing that the sum of two connected sets in  has non-empty interior under weaker conditions, specifically when only one set is compact and not a line segment.

Findings

Sum of connected sets with one non-line segment compact set has non-empty interior.

Improves previous results by relaxing compactness assumptions.

Results applicable in  for connected sets with cardinality greater than one.

Abstract

We prove that for two connected sets $E,F\subset\mathbb{R}^2$ with cardinalities greater than $1$, if one of $E$ and $F$ is compact and not a line segment, then the arithmetic sum $E+F$ has non-empty interior. This improves a recent result of Banakh, Jab{\l}o\'nska and Jab{\l}o\'nski [4,Theorem 4] in dimension two by relaxing their assumption that $E$ and $F$ are both compact.