Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points with mutual integer distances on the shortest line containing points in $\mathcal{S}$ satisfies the lower bound \begin{align} \gg_n \sqrt{n}|\mathcal{S}\bigcap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]|\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}\cap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}\\k>1}}\frac{1}{k},\nonumber \end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.