<A HREF="https://openreview.net/group?id=ICLR.cc/2021/Conference">ICLR 2021</A> has contributions that could qualify as "rigorous results", one you may like is <A HREF="https://openreview.net/forum?id=O-XJwyoIF-k">Minimum Width for Universal Approximation.</A>

> The universal approximation property of width-bounded networks is
> characterized in terms of the input dimension  $d_x$ and the output
> dimension $d_y$. The minimum width required for the universal
> approximation of $L^p$ functions is exactly $\max \{d_x+1,d_y\}.$