The homogeneous Sobolev spaces $\dot{W}^{s,n/s}(\mathbb{R}^n)$ all fail to embed into $L^{\infty}$ when $0<s<n$, but only barely so. In dimension $n \geq 2$, Bourgain and Brezis found a rather useful remedy for this failure when $s = 1$, by considering how well a general $\dot{W}^{1,n}$ function can be approximated by an $L^{\infty}$ function on $\mathbb{R}^n$. In this talk we give an extension of their results to $\dot{W}^{s,n/s}$ for all $0 < s < n$. This is joint work with Pierre Bousquet, Emmanuel Russ and Yi Wang.