Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that $$x^*=|x|^{-2}x.$$ For a set $E$, we set $E^*=\{x^*:x\in E, x\not=0\} $. We know that a set $E\subset\mathbb{R}^m$ is thin at $x\not=0$, if and only if $E^*$ is thin at $x^*$. We know also that $\infty$ is an irregular boundary point of any unbounded open set $\Omega$ in $\mathbb{R}^m$ if $m=2$ (we suppose, if $m=2$, that the complement of $\Omega$ is non polar, i.e. $\Omega$ is Greenian), but not if $m>2$. According to a classical theorem for $m>2$, $\infty$ is negligible for $\Omega$, if and only if $\mathbb{R}^m\setminus\Omega$ is non-thin at $\infty$ (see Armitage and Gardiner "Classical Potential Theory" pg 214 - 15).
My question is: What is a simple example of such open set? That is, an open set $\Omega$ for $m>2$ such that $\mathbb{R}^m\setminus\Omega$ is non-thin at $\infty$?
