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

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, Jabłońska and Jabłoński [4,Theorem 4] in dimension two by relaxing their assumption that $E$ and $F$ are both compact.